env['ASYMPTOTE'] = 'asy'
env['ASYMPTOTEFLAGS'] = SCons.Util.CLVar(
'-tex pdflatex -inlineimage -inlinetex')
- env['ASYMPTOTECOM'] = 'cd ${TARGET.dir} && $ASYMPTOTE $ASYMPTOTEFLAGS ${SOURCE.filebase}'
+ env['ASYMPTOTECOM'] = 'cd ${TARGET.dir}' ## && $ASYMPTOTE $ASYMPTOTEFLAGS ${SOURCE.filebase}'
env.Append(SCANNERS=SCons.Scanner.Base(
function=asymptote_scan,
name='Asymptote',
identical.\label{fig:plasmid}}
\end{figure}
+\index{\species{Escherichia coli}}
The plasmid is then transformed into the host, usually
\species{Escherichia coli}\citep{carrion-vazquez99b,bartels03,ma10} or
a proprietary equivalent such as Agilent's SURE 2 Supercompetent
--- /dev/null
+\chapter{The Gumbel (minimum) distribution}
+\label{sec:gumbel}
+
+The Gumbel (minimum) distribution is also refered to as the extreme
+value type I distribution distribution\citep{NIST:gumbel}, the
+log-Weibull distribution\citep{wikipedia:gumbel}, the double
+exponential distribution\citep{wikipedia:gumbel}. The Gumbel
+distribution is often confused with the Gompertz
+distribution\citep{wikipedia:gumbel}. To avoid confusion, we can
+express both Gumbel and Gompertz in terms of the Generalized extreme
+value distribution (GEV, a.k.a. the Fisher--Tippett
+distribution\citep{wikipedia:GEV}), which has a probability
+distribution given by
+
+\begin{align}
+ P_\text{GEV}(x|\mu,\sigma,\eta)
+ &= \frac{1}{\sigma}t(x)\exp\p({-t(x)}) \\
+ t(x) &= \begin{cases}
+ \p({1 + \p({\frac{x-\mu}{\sigma}})\eta})^{-1/\eta} &
+ \text{if } \eta \ne 0 \\
+ \exp\p({-\frac{x-\mu}{\sigma}}) & \text{if } \eta = 0
+ \end{cases}
+\end{align}
+where $\mu\in\Reals$ is the location parameter, $\sigma>0$ is the
+scale parameter, and $\eta\in\Reals$ is the shape
+parameter\citep{wikipedia:GEV}.
+
+To recover the Gumbel distribution, set $\eta=0$.
+\begin{align}
+ P_\text{Gumbel}(x|\mu,\sigma)
+ &= \frac{1}{\sigma}\exp\p({-\frac{x-\mu}{\sigma}})
+ \exp\p({\exp\p({-\frac{x-\mu}{\sigma}})}) \\
+ &= \frac{1}{\sigma}\exp\p({z - \exp(-z)}) \;,
+\end{align}
+where $z\equiv (x-\mu)/\sigma$. This form matches
+\citet{wikipedia:gumbel} and, with the replacements
+$\mu\rightarrow\alpha$ and $\sigma\rightarrow\rho$, also matches
+\cref{eq:sawsim:gumbel}.
+% TODO: add gumbel58 citations where appropriate
+
+To recover the Gompertz distribution\citet{wikipedia:gompertz}
+\begin{align}
+ P_\text{Gompertz}(x|\nu,b)
+ &= b\nu\exp(bx)\exp(\nu)\exp(-\nu\exp(bx))
+\end{align}
+, set $$.
+
+Finally, there are a few other similarly named distributions to watch
+out for. The Type-1 Gumbel distribution\citet{wikipedia:gumbel-t1}
+\begin{equation}
+ P_\text{Type-1 Gumbel}(x|a,b)
+ = ab\exp\p({-(b\exp(-ax)+ax)})
+ = -\exp(-b)\cdotP_\text{Gompertz})(x|b,-a)
+\end{equation}
+is similar to the Gompertz distrubution, differing only by a constant
+scale factor. Since both probability distributions are normalized,
+the difference comes from the range of allowed $x$. For the Gompertz
+distribution, $x\ge0$. For the Type-1 Gumbel distribution,
+$-\infty<x<\infty$. Note that the Type-1 Gumbel distribution is not
+the same as the Gumbel (minimum) distribution.
+
+The Type-2 Gumbel distribution\citet{wikipedia:gumbel-t2}
+\begin{equation}
+ P_\text{Type-2 Gumbel}(x|a,b) = abx^{-a-1}\exp\p({-bx^{-a}})
+\end{equation}
+has $x$ being raised to powers (vs. $e$ being raised to powers in the
+other distributions), so it is an entirely different beast.
+
+Properties of the Bell model recieve more coverage under the name of
+the older and equivalent Gompertz
+distribution\citep{gompertz25,olshansky97,wu04}. eq:sawsim:order-depA
+warning about the ``Gompertz'' model is in order, because there seem
+to be at least two unfolding/dying rate formulas that go by that name.
+Compare, for example, \citet{braverman08} Eqn.~5 and \citet{juckett93}
+Fig.~2.},
\begin{figure}
\asyinclude{figures/schematic/landscape-cant}
- \caption{Energy landscape schematic.\label{fig:landscape}}
+ \caption{Energy landscape of a protein being stretched by an
+ external force. For a discussion of methods for calculating
+ unfolding rates from such a landscape, see
+ \cref{sec:sawsim:rate}.\label{fig:landscape:cantilever}}
\end{figure}
The presence of attached linkers and cantilevers alters the free
and stiffer linkers will increase the mean unfolding force.
Unfolded I27 domains can be well-modeled as wormlike chains (WLCs,
-\cref{sec:tension:wlc})\citep{carrion-vazquez99b}, where $p \approx
-4\U{\AA}$ is the persistence length, and $L \approx 28\U{nm}$ is the
-contour length of the unfolded domain. Obviously effective stiffness
-of an unfolded I27 domain is highly dependent on the unfolding force,
-and for tensions $\sim 280\U{pN}$ is $\sim 190\U{pN/nm}$. This is
-within a factor of four of common cantilever spring constants, so
-cantilever stiffness drives the effective spring constant for the
-first four domains, after which point I27 stiffness takes the lead.
+\cref{sec:sawsim:tension:wlc})\citep{carrion-vazquez99b}, where $p
+\approx 4\U{\AA}$ is the persistence length, and $L \approx 28\U{nm}$
+is the contour length of the unfolded domain. Obviously effective
+stiffness of an unfolded I27 domain is highly dependent on the
+unfolding force, and for tensions $\sim 280\U{pN}$ is $\sim
+190\U{pN/nm}$. This is within a factor of four of common cantilever
+spring constants, so cantilever stiffness drives the effective spring
+constant for the first four domains, after which point I27 stiffness
+takes the lead.
\begin{figure}
\begin{tikzpicture}
% Inspired by Florian Hollandt's RNA codons
import wtk_graph;
-size(15cm,10cm,IgnoreAspect);
+size(8cm,6cm,IgnoreAspect);
scale(Log, Linear, Log);
real xscale=1;
real yscale=1e9;
graphMatrixFile("data", xscale=xscale, yscale=yscale,
- x_label="$k_{uo}$ (s$^{-1}$)", y_label="$x_u$ (nm)",
+ x_label="$k_{uo}$ (s$^{-1}$)", y_label="$\Delta x_u$ (nm)",
palette_label="$D_\text{JS}$");
label(sLabel("Fit quality"), point(N),N);
}
xlimits(0, 300e-9*xscale, crop=true);
-xaxis(sLabel("Distance (nm)"), BottomTop, LeftTicks);
+xaxis(sLabel("Distance (nm) ($x_t-x_c$)"), BottomTop, LeftTicks);
yaxis(sLabel("Force (pN)"), LeftRight, RightTicks);
label(sLabel("Simulated force curves"), point(N), N);
from site_cons.site_init import link_wtk_graph
FIGURES = ['unfolding', 'afm', 'kramers-integrand', 'landscape',
- 'landscape-cant', 'monte-carlo', 'piezo']
+ 'landscape-bell', 'landscape-cant', 'monte-carlo', 'piezo',
+ 'wlc-extension', 'wlc-model', 'fjc-extension', 'fjc-model']
# Get the passed in environment.
Import('env')
deps = [wtk_graph]
if fig in ['unfolding', 'afm']:
deps.append('base_afm.asy')
+ if fig in ['unfolding', 'wlc-model', 'fjc-model']:
+ deps.append('wiggle')
env.Asymptote(['%s.asy' % fig] + deps)
# Pass back the modified environment.
--- /dev/null
+import wtk_graph;
+
+size(6cm, 4cm, IgnoreAspect);
+
+scale(Linear, Linear);
+
+real n = 100;
+real x_min = 0;
+real x_max = 0.8;
+
+struct inverse_langevin {
+ real y;
+
+ void operator init(real y=0) {
+ this.y = y;
+ }
+
+ real coth(real x) { // hyperbolic cotangent
+ real e = exp(2*x);
+ return (e + 1)/(e - 1);
+ }
+
+ real langevin(real x) {
+ if (x == 0)
+ return 0;
+ return coth(x) - 1/x;
+ }
+
+ real offset_langevin(real x) {
+ return langevin(x) - this.y;
+ }
+
+ real langevin_prime(real x) {
+ if (x == 0)
+ return 1/3;
+ return 1 - coth(x)^2 + 1/x^2;
+ }
+
+ real inv_langevin(real x) {
+ this.y = x;
+ return newton(this.offset_langevin, langevin_prime, x1=0, x2=10);
+ }
+}
+
+inverse_langevin inv_langevin = inverse_langevin();
+
+real fjc(real x) {return inv_langevin.inv_langevin(x);}
+real wlc(real x) {return 0.25*(1/(1-x)^2 - 1) + x;}
+
+draw(graph(fjc, x_min, x_max, operator ..), red);
+draw(graph(wlc, x_min, x_max, operator ..), dashed + blue);
+
+label(sLabel("Freely-jointed chain extension"), point(N), N);
+xaxis(sLabel("End-to-end distance $x/Nl$"), BottomTop, LeftTicks);
+yaxis(sLabel("Tension $\frac{Fl}{k_B T}$"), LeftRight, RightTicks);
--- /dev/null
+import wiggle; // for RandomWalk
+
+RandomWalk w = RandomWalk(dx=1cm);
+w.align(90);
+guide g = w.straight_guide();
+draw(g, red);
+dot(g, linewidth(2pt) + blue);
+dot(w.start);
+dot(w.end);
--- /dev/null
+import wtk_graph;
+
+size(6cm, 4cm, IgnoreAspect);
+
+scale(Linear, Linear);
+
+real x_min = 0;
+real x_max = 1;
+real f_min = 0; // zoom in on region of interest
+real f_max = 1;
+pair folded = (0.2, 0.2);
+pair transition = (0.8, 0.8);
+pair forced_transition = (0.8, 0.5);
+real dx = 0.1; // width of energy level lines
+
+void draw_level(pair center, Label label, pen p=currentpen) {
+ draw(Scale(center-(dx/2,0)) -- Scale(center+(dx/2,0)), p);
+ dot(label, Scale(center));
+}
+
+void draw_rate(pair a, pair b, Label label, pen p=currentpen) {
+ path arrow_path = Scale(a){dir(70)} .. {dir(-60)}Scale(b);
+ draw(arrow_path, p, Arrows, Margin(1, 1));
+ label(label, point(arrow_path, length(arrow_path)/2));
+}
+
+draw_level(folded, Label("folded", align=S));
+draw_level(transition, Label("transition", align=S));
+draw_level(forced_transition, Label(
+ minipage("\center{forced transition}", 60), align=S), dashed);
+
+draw_rate(folded, transition, Label("$k_{u0}$", align=NW));
+draw_rate(folded, forced_transition, Label("$k_{uF}$", align=SE), dashed);
+
+xlimits(x_min, x_max, Crop);
+ylimits(f_min, f_max, Crop);
+
+label(sLabel("Bell model unfolding"), point(N), N);
+xaxis(sLabel("End-to-end distance $x$"), BottomTop);
+yaxis(sLabel("Free energy $U_F$"), LeftRight);
//size(x=9cm,keepAspect=true);
-import stats; // for unitrand
import base_afm; // for Cantilever, Substrate, ...
-
-struct Wiggle {
- // Generate a horizontal wiggly line, centered at CENTER (x,y)
- // with a width WIDTH x
- // and randomly distributed heights.
- // The random points are unformly distributed within an envelope
- // with the maximum possible deviation given by ROUGHNESS x
-
- pair start;
- pair end;
- real roughness;
- real tense;
- int num;
- int random_seed;
- pair points[];
-
- void generate(bool labels=false) {
- int n = this.num;
- pair step = (this.end-this.start) / (n-1);
- pair perp = scale(this.roughness)*rotate(90)*unit(step);
- real y, envelope;
- srand(this.random_seed);
- for (int i=0; i < n; i+=1) {
- if (i == (n-1)/2 || i == 0 || i == n-1) {
- y = 0.0;
- } else {
- real frac = i/n;
- envelope = (frac*(1.0-frac)*4)**3;
- y = (2*unitrand()-1.0) * envelope;
- }
- this.points.push(this.start + i*step + perp*y);
- if (labels==true)
- dot(format("%d", i), this.points[-1], S);
- }
- }
-
- guide guide() {
- guide g;
- for (int i; i < this.points.length; i+=1) {
- g = g .. tension this.tense .. this.points[i];
- }
- this.end = point(g, length(g));
- return g;
- }
-
- void operator init(pair start=(0,0), pair end=(1cm,0), real roughness=1cm,
- real tense=1, int num=15, int random_seed=5) {
- this.start = start;
- this.end = end;
- this.roughness = roughness;
- this.tense = tense;
- this.num = num;
- this.random_seed = random_seed;
- this.generate();
- }
-}
+import wiggle; // for Wiggle
struct ProteinChain {
// Generate and draw a mixed protein (2nd protein unfolded)
--- /dev/null
+// Define Wiggle structure for drawing random wiggles.
+
+struct Wiggle {
+ // Generate a horizontal wiggly line, centered at CENTER (x,y)
+ // with a width WIDTH x
+ // and randomly distributed heights.
+ // The random points are unformly distributed within an envelope
+ // with the maximum possible deviation given by ROUGHNESS x
+
+ pair start;
+ pair end;
+ real roughness;
+ real tense;
+ int num;
+ int random_seed;
+ pair points[];
+
+ void generate(bool labels=false) {
+ int n = this.num;
+ pair step = (this.end-this.start) / (n-1);
+ pair perp = scale(this.roughness)*rotate(90)*unit(step);
+ real y, envelope;
+ srand(this.random_seed);
+ for (int i=0; i < n; i+=1) {
+ if (i == (n-1)/2 || i == 0 || i == n-1) {
+ y = 0.0;
+ } else {
+ real frac = i/n;
+ envelope = (frac*(1.0-frac)*4)**3;
+ y = (2*unitrand()-1.0) * envelope;
+ }
+ this.points.push(this.start + i*step + perp*y);
+ if (labels==true)
+ dot(format("%d", i), this.points[-1], S);
+ }
+ }
+
+ guide guide() {
+ guide g;
+ for (int i=0; i < this.points.length; i+=1) {
+ g = g .. tension this.tense .. this.points[i];
+ }
+ this.end = point(g, length(g));
+ return g;
+ }
+
+ void operator init(pair start=(0,0), pair end=(1cm,0), real roughness=1cm,
+ real tense=1, int num=15, int random_seed=5) {
+ this.start = start;
+ this.end = end;
+ this.roughness = roughness;
+ this.tense = tense;
+ this.num = num;
+ this.random_seed = random_seed;
+ this.generate();
+ }
+}
+
+struct RandomWalk {
+ // Generate a random walk starting from START (x,y)
+ // with NUM steps, each of step-size DX,
+ // and randomly distributed step directions.
+
+ pair start;
+ pair end;
+ real dx;
+ int num;
+ real tense;
+ int random_seed;
+ pair points[];
+
+ void generate(bool labels=false) {
+ int n = this.num;
+ srand(this.random_seed);
+ pair p = this.start;
+ this.points.push(p);
+ if (labels==true)
+ dot("0", this.points[0], S);
+ for (int i=1; i <= this.num; i+=1) {
+ real theta = 360*unitrand();
+ p += rotate(theta)*(this.dx, 0);
+ this.points.push(p);
+ if (labels==true)
+ dot(format("%d", i), this.points[i], S);
+ }
+ this.end = p;
+ }
+
+ void align(real dir) {
+ real dtheta = dir - degrees(this.end - this.start);
+ for (int i=1; i < this.points.length; i+=1) {
+ this.points[i] = rotate(dtheta, this.start)*this.points[i];
+ }
+ this.end = this.points[this.num];
+ }
+
+ guide curved_guide() {
+ guide g;
+ for (int i=0; i < this.points.length; i+=1) {
+ g = g .. tension this.tense .. this.points[i];
+ }
+ return g;
+ }
+
+ guide straight_guide(interpolate join=operator ..) {
+ guide g;
+ for (int i=0; i < this.points.length; i+=1) {
+ g = g -- this.points[i];
+ }
+ return g;
+ }
+
+ void operator init(pair start=(0,0), real dx=3mm,
+ real tense=1, int num=15, int random_seed=25) {
+ this.start = start;
+ this.dx = dx;
+ this.tense = tense;
+ this.num = num;
+ this.random_seed = random_seed;
+ this.generate();
+ }
+}
--- /dev/null
+import wtk_graph;
+
+size(6cm, 4cm, IgnoreAspect);
+
+scale(Linear, Linear);
+
+real x_min = 0;
+real x_max = 0.8;
+
+real wlc(real x) {return 0.25*(1/(1-x)^2 - 1) + x;}
+
+draw(graph(wlc, x_min, x_max, operator ..), red);
+
+label(sLabel("Wormlike chain extension"), point(N), N);
+xaxis(sLabel("End-to-end distance $x/L$"), BottomTop, LeftTicks);
+yaxis(sLabel("Tension $\frac{Fp}{k_B T}$"), LeftRight, RightTicks);
--- /dev/null
+import wiggle; // for RandomWalk
+
+RandomWalk w = RandomWalk(dx=1cm);
+w.align(90);
+draw(w.curved_guide(), red);
+dot(w.start);
+dot(w.end);
import wtk_graph;
-size(15cm, 10cm, IgnoreAspect);
+size(6cm, 4cm, IgnoreAspect);
scale(Linear, Log);
real fscale=1e12;
import wtk_graph;
-size(15cm,10cm,IgnoreAspect);
+size(6cm,4cm,IgnoreAspect);
scale(Linear, Linear);
real k = 0.05; /* spring constant in N/m */
graphSawtooth("fig3.dat", k=k, xscale=xscale, fscale=fscale,
df=400e-12, p=phard);
-xaxis(sLabel("Distance (nm)"), BottomTop, LeftTicks);
+xaxis(sLabel("Distance (nm) ($x_t-x_c$)"), BottomTop, LeftTicks);
yaxis(sLabel("Force (pN)"), LeftRight, RightTicks);
label(sLabel("Simulated force curves"), point(N), N);
experimental data remains elusive. For example, experimental pulling
speeds are on the order of \bareU{$\mu$m/s}, while simulation pulling
speeds are on the order of
-\bareU{m/s}\citep{lu98,lu99,zhao06,berkemeier11}.
+\bareU{m/s}\citep{lu98,lu99,rief02,zhao06,berkemeier11}.
% why AFM & what an AFM is
Single molecule techniques for manipulating biopolymers include
% Fourier Transform to frequency space
\newcommand{\Fourf}[1]{\ensuremath{{\mathcal F}_f\left\{ {#1} \right\}}}
+% Symbol denoting the Langevin function
+\newcommand{\Langevin}{\ensuremath{\mathcal{L}}}
+
% Integral from #1 to #2 of #4 with respect to #3
\newcommand{\integral}[4]{\ensuremath{\int_{#1}^{#2} {#4} \dd{#3}}}
% Integral from -infty to +infty of #2 with respect to #1
\newcommand{\colC}[1]{\textcolor{green}{#1}}
\newcommand{\ie}{\emph{i.e.}} % "id est" or "in other words"
+\newcommand{\insilico}{\emph{in silico}} % quasi latin for "on a computer"
+\newcommand{\invitro}{\emph{in vitro}} % latin for "in glass"
+\newcommand{\invivo}{\emph{in vivo}} % latin for "in living organisms"
\newcommand{\ensuretext}[1]{\ensuremath{\text{#1}}}
% Hyeon and Thirumalai equation number #1
%\newcommand{\avg}[1]{\ensuremath{\left\langle {#1} \right\rangle}}
\newcommand{\logp}[1]{\ensuremath{\log\!\!\left( {#1} \right)}}
% \! is a negative thin space to get the paren closer to the log
-\renewcommand{\r}{\ensuremath{r_f}}
-\newcommand{\rs}[1]{\ensuremath{r_{f{#1}}}}% to avoid double-subscripting
+%\renewcommand{\r}{\ensuremath{r_f}}
+%\newcommand{\rs}[1]{\ensuremath{r_{f{#1}}}}% to avoid double-subscripting
\newcommand{\ep}{\varepsilon}
\newcommand{\species}[1]{\emph{#1}} % \species{Homo sapiens}
\newcommand{\Hooke}{\citetalias{sandal09}}
\newcommand{\calibcant}{\citetalias{calibcant}}
\newcommand{\pyafm}{\citetalias{pyafm}}
+\newcommand{\pysawsim}{\texttt{pysawsim}}
\newcommand{\sawsim}{\citetalias{king10}}
\newcommand{\stepper}{\citetalias{stepper}}
\usepackage{amssymb}
% fonts like mathbb (blackboard bold) and mathcal (caligraphic)
\usepackage{amsfonts}
-% double struct math fonts
+% double struck math fonts
\usepackage{dsfont}
+% nicer table formatting (\toprule, \cimidrule, \midrule, and \bottomrule)
+\usepackage{booktabs}
\usepackage{xcolor}
% for the whole article with \linenumbers after \end{preamble}.
% Note that line numbering messes with TOC/LOT/LOF \numberline,
% so ensure line numbering is off for the tables.
-\usepackage{lineno}
+\iffinal{}{\usepackage{lineno}}
\usepackage{pgf} % fancy graphics
\usepackage{tikz} % a nice, inline PGF frontend
\usepackage{epsdice} % dice-face font
+% nicer verbatim environments (Verbatim)
+\usepackage{fancyvrb}
+
+\RecustomVerbatimEnvironment{Verbatim}{Verbatim}{%
+ frame=lines,commandchars=\\\{\}}
+
\usepackage{wtk_cmmds} % common personal macros, in ~/texmf/tex/latex/
\input{local_cmmds}
@string{MAllen = "Allen, Mark D."}
@string{RAlon = "Alon, Ronen"}
@string{PAmanatides = "Amanatides, P."}
+@string{NMAmer = "Amer, Nabil M."}
@string{AJP = "American Journal of Physics"}
@string{APS = "American Physical Society"}
@string{ASA = "American Statistical Association"}
@string{BBagchi = "Bagchi, B."}
@string{MBalamurali = "Balamurali, M. M."}
@string{DBaldwin = "Baldwin, D."}
+@string{ABaljon = "Baljon, Arlette R. C."}
+@string{RBallerini = "Ballerini, R."}
@string{RMBallew = "Ballew, R. M."}
@string{MBalsera = "Balsera, M."}
@string{GBaneyx = "Baneyx, Gretchen"}
@string{NBhasin = "Bhasin, Nishant"}
@string{KBiddick = "Biddick, K."}
@string{KBillings = "Billings, Kate S."}
+@string{GBinnig = "Binnig, Gerd"}
@string{BCBPRC = "Biochemical and Biophysical Research Communications"}
@string{Biochem = "Biochemistry"}
@string{BBABE = "Biochimica et Biophysica Acta (BBA) - Bioenergetics"}
-@string{BIOINFO = "Bioinformatics (Oxford, England)",}
+@string{BIOINFO = "Bioinformatics (Oxford, England)"}
@string{BPJ = "Biophysical Journal"}
-%@string{BPJ = "Biophys. J."}
+%string{BPJ = "Biophys. J."}
@string{BIOSENSE = "Biosensors and Bioelectronics"}
@string{BIOTECH = "Biotechnology and Bioengineering"}
@string{JBirchler = "Birchler, James A."}
@string{ICampbell = "Campbell, Iain D."}
@string{MJCampbell = "Campbell, M. J."}
@string{YCao = "Cao, Yi"}
+@string{MCapitanio = "Capitanio, M."}
@string{MCargill = "Cargill, M."}
@string{PCarl = "Carl, Philippe"}
@string{BACarnes = "Carnes, B. A."}
@string{CCecconi = "Cecconi, Ciro"}
@string{ACenter = "Center, A."}
@string{HSChan = "Chan, H. S."}
+@string{AChand = "Chand, Ami"}
@string{IChandramouliswaran = "Chandramouliswaran, I."}
@string{CHChang = "Chang, Chung-Hung"}
@string{EChapman = "Chapman, Edwin R."}
@string{KClerc-Blankenburg = "Clerc-Blankenburg, K."}
@string{NJCobb = "Cobb, Nathan J."}
@string{FSCollins = "Collins, Francis S."}
+@string{CUP = "Columbia University Press"}
@string{CPR = "Computer Physics Reports"}
@string{UniProtConsort = "Consortium, The UniProt"}
@string{MConti = "Conti, Matteo"}
@string{TDrobek = "Drobek, T."}
@string{OKDudko = "Dudko, Olga K."}
@string{ADunham = "Dunham, A."}
+@string{DDunlap = "Dunlap, D."}
@string{PDunn = "Dunn, P."}
@string{EMBORep = "EMBO Rep"}
@string{EMBO = "EMBO Rep."}
@string{NSGavrilova = "Gavrilova, N. S."}
@string{WGe = "Ge, W."}
@string{GENE = "Gene"}
+@string{CGerber = "Gerber, Christoph"}
@string{CGergely = "Gergely, C."}
@string{RGibbs = "Gibbs, R."}
@string{DGilbert = "Gilbert, D."}
@string{HGire = "Gire, H."}
+@string{MGiuntini = "Giuntini, M."}
@string{SGlanowski = "Glanowski, S."}
@string{JGlaser = "Glaser, Jens"}
@string{KGlasser = "Glasser, K."}
@string{ZGu = "Gu, Z."}
@string{PGuan = "Guan, P."}
@string{RGuigo = "Guig\'o, R."}
+@string{EJGumbel = "Gumbel, Emil Julius"}
@string{HJGuntherodt = "Guntherodt, Hans-Joachim"}
@string{NGuo = "Guo, N."}
@string{YGuo = "Guo, Yi"}
@string{JHemmerle = "Hemmerle, J."}
@string{SHenderson = "Henderson, S."}
@string{BHeymann = "Heymann, Berthold"}
-@string{NHiaro = "Hiaro, N."
+@string{NHiaro = "Hiaro, N."}
@string{MEHiggins = "Higgins, M. E."}
@string{LHillier = "Hillier, L."}
@string{HHinssen = "Hinssen, Horst"}
@string{MIvemeyer = "Ivemeyer, M."}
@string{DIzhaky = "Izhaky, David"}
@string{SIzrailev = "Izrailev, S."}
-%@string{JACS = "J Am Chem Soc"}
+%string{JACS = "J Am Chem Soc"}
@string{JACS = "Journal of the American Chemical Society"}
+@string{JAP = "Journal of Applied Physics"}
@string{JBM = "J Biomech"}
@string{JBT = "J Biotechnol"}
@string{JEChem = "Journal of Electroanalytical Chemistry"}
@string{JMathBiol = "J Math Biol"}
+@string{JPhysio = "Journal of physiology"}
@string{JStructBiol = "Journal of structural biology"}
@string{JTB = "J Theor Biol"}
@string{WJang = "Jang, W."}
@string{PLelkes = "Lelkes, Peter I."}
@string{OLequin = "Lequin, Olivier"}
@string{CLethias = "Lethias, Claire"}
+@string{SLeuba = "Leuba, Sanford H."}
@string{ALeung = "Leung, A."}
@string{AJLevine = "Levine, A. J."}
@string{CLevinthal = "Levinthal, Cyrus"}
@string{SHLin = "Lin, Sheng-Hsien"}
@string{XLin = "Lin, X."}
@string{JLindahl = "Lindahl, Joakim"}
+@string{SLindsay = "Lindsay, Stuart M."}
@string{WALinke = "Linke, Wolfgang A."}
@string{RLippert = "Lippert, R."}
@string{JLis = "Lis, John T."}
@string{JFMarko = "Marko, John F."}
@string{MMarra = "Marra, M."}
@string{PMarszalek = "Marszalek, Piotr E."}
+@string{MMartin = "Martin, M. J."}
@string{HMassa = "Massa, H."}
@string{JMathe = "Math\'e, J\'er\^ome"}
@string{AMatouschek = "Matouschek, Andreas"}
@string{GVMerkulov = "Merkulov, G. V."}
@string{HMetiu = "Metiu, Horia"}
@string{NMetropolis = "Metropolis, Nicholas"}
+@string{GMeyer = "Meyer, Gerhard"}
@string{HMi = "Mi, H."}
@string{MMickler = "Mickler, Moritz"}
@string{AMiller = "Miller, A."}
@string{VParpura = "Parpura, Vladimir"}
@string{APastore = "Pastore, A."}
@string{APatrinos = "Patrinos, Aristides"}
+@string{FPavone = "Pavone, F. S."}
@string{SHPayne = "Payne, Stephen H."}
@string{JPeck = "Peck, J."}
@string{HPeng = "Peng, Haibo"}
@string{PA = "Pflugers Arch"}
@string{PTRSL = "Philosophical Transactions of the Royal Society of London"}
@string{PR:E = "Phys Rev E Stat Nonlin Soft Matter Phys"}
-@string{PRL = "Phys Rev Lett"}
+@string{PRL = "Physical review letters"}
+%string{PRL = "Phys Rev Lett"}
@string{Physica = "Physica"}
@string{GPing = "Ping, Guanghui"}
@string{NPinotsis = "Pinotsis, Nikos"}
@string{WPyckhout-Hintzen = "Pyckhout-Hintzen, Wim"}
@string{SQin = "Qin, S."}
@string{SRQuake = "Quake, Stephen R."}
+@string{CQuate = "Quate, Calvin F."}
@string{HQureshi = "Qureshi, H."}
@string{SERadford = "Radford, Sheena E."}
@string{MRaible = "Raible, M."}
@string{FRief = "Rief, Frederick"}
@string{MRief = "Rief, Matthias"}
@string{KRitchie = "Ritchie, K."}
+@string{MRobbins = "Robbins, Mark O."}
@string{RJRoberts = "Roberts, R. J."}
@string{RRobertson = "Robertson, Ragan B."}
@string{HRoder = "Roder, Heinrich"}
@string{YHRogers = "Rogers, Y. H."}
@string{SRogic = "Rogic, S."}
@string{MRoman = "Roman, Marisa"}
+@string{GRomano = "Romano, G."}
@string{DRomblad = "Romblad, D."}
@string{RRos = "Ros, Robert"}
@string{BRosenberg = "Rosenberg, B."}
@string{TSato = "Sato, Takehiro"}
@string{PSchaaf = "Schaaf, P."}
@string{RSchafer = "Schafer, Rolf"}
+@string{TESchafer = "Sch{\"a}fer, Tilman E."}
@string{NScherer = "Scherer, Norbert F."}
@string{SScherer = "Scherer, S."}
@string{MSchilhabel = "Schilhabel, M."}
@string{PSeranski = "Seranski, P."}
@string{RSesboue = {Sesbo\"u\'e, R.}}
@string{EShakhnovich = "Shakhnovich, Eugene"}
-@string{GShan, "Shan, Guiye"}
+@string{GShan = "Shan, Guiye"}
@string{JShang = "Shang, J."}
@string{WShao = "Shao, W."}
@string{DSharma = "Sharma, Deepak"}
@string{PHVerdier = "Verdier, Peter H."}
@string{IVetter = "Vetter, Ingrid R."}
@string{WVetterling = "Vetterling, W."}
+@string{MViani = "Viani, Mario B."}
@string{JCVoegel = "Voegel, J.-C."}
@string{VVogel = "Vogel, Viola"}
@string{CWagner-McPherson = "Wagner-McPherson, C."}
@string{MWen = "Wen, M."}
@string{JWetter = "Wetter, J."}
@string{AWhittaker = "Whittaker, A."}
+@string{YWickramasinghe = "Wickramasinghe, H. K."}
@string{RWides = "Wides, R."}
@string{AWiita = "Wiita, Arun P."}
@string{MWilchek = "Wilchek, Meir"}
@string{AWilcox = "Wilcox, Alexander J."}
@string{Williams = "Williams"}
+@string{CCWilliams = "Williams, C. C."}
@string{MWilliams = "Williams, M."}
@string{SWilliams = "Williams, S."}
-@strinf{MWilmanns = "Wilmanns, Matthias"}
+@string{MWilmanns = "Wilmanns, Matthias"}
@string{RKWilson = "Wilson, R. K."}
@string{SWilson = "Wilson, Scott"}
@string{SWindsor = "Windsor, S."}
@string{WZhuang = "Zhuang, Wei"}
@string{NZinder = "Zinder, N."}
@string{RCZinober = "Zinober, Rebecca C."}
+@string{JZlatanova = "Zlatanova, Jordanka"}
@string{PZou = "Zou, Peng"}
@string{RZwanzig = "Zwanzig, R."}
@string{arXiv = "arXiv"}
url = "http://www.itl.nist.gov/div898/handbook/eda/section3/eda366g.htm"
}
+@misc{ wikipedia:gumbel,
+ author = "Wikipedia",
+ title = "Gumbel distribution --- {W}ikipedia{,} The Free Encyclopedia",
+ year = 2012,
+ url = "http://en.wikipedia.org/wiki/Gumbel_distribution",
+}
+
+@book { gumbel58,
+ author = EJGumbel,
+ title = "Statistics of Extremes",
+ year = 1958,
+ publisher = CUP,
+ address = "New York",
+ note = "TODO: read",
+}
+
+@misc{ wikipedia:GEV,
+ author = "Wikipedia",
+ title = "Generalized extreme value distribution --- {W}ikipedia{,}
+ The Free Encyclopedia",
+ year = 2012,
+ url = "http://en.wikipedia.org/wiki/Generalized_extreme_value_distribution",
+}
+
+@misc{ wikipedia:gompertz,
+ author = "Wikipedia",
+ title = "Gompertz distribution --- {W}ikipedia{,} The Free Encyclopedia",
+ year = 2012,
+ url = "http://en.wikipedia.org/wiki/Gompertz_distribution",
+}
+
+@misc{ wikipedia:gumbel-t1,
+ author = "Wikipedia",
+ title = "Type-1 Gumbel distribution --- {W}ikipedia{,} The Free
+ Encyclopedia",
+ year = 2012,
+ url = "http://en.wikipedia.org/wiki/Type-1_Gumbel_distribution",
+}
+
+@misc{ wikipedia:gumbel-t2,
+ author = "Wikipedia",
+ title = "Type-2 Gumbel distribution --- {W}ikipedia{,} The Free
+ Encyclopedia",
+ year = 2012,
+ url = "http://en.wikipedia.org/wiki/Type-2_Gumbel_distribution",
+}
+
@article { allemand03,
author = JFAllemand #" and "# DBensimon #" and "# VCroquette,
title = "Stretching {DNA} and {RNA} to probe their interactions with
microscopic theories in reproducing the experimental data."
}
+@article{ dudko08,
+ author = OKDudko #" and "# GHummer #" and "# ASzabo,
+ title = "Theory, analysis, and interpretation of single-molecule
+ force spectroscopy experiments.",
+ journal = PNAS,
+ year = 2008,
+ month = oct,
+ day = 14,
+ address = "Department of Physics and Center for Theoretical
+ Biological Physics, University of California at San Diego, La
+ Jolla, CA 92093, USA.
+ dudko@physics.ucsd.edu",
+ volume = 105,
+ number = 41,
+ pages = "15755--15760",
+ keywords = "DNA",
+ keywords = "Half-Life",
+ keywords = "Kinetics",
+ keywords = "Microscopy, Atomic Force",
+ keywords = "Motion",
+ keywords = "Nucleic Acid Conformation",
+ keywords = "Nucleic Acid Denaturation",
+ keywords = "Protein Folding",
+ keywords = "Thermodynamics",
+ abstract = "Dynamic force spectroscopy probes the kinetic and
+ thermodynamic properties of single molecules and molecular
+ assemblies. Here, we propose a simple procedure to extract kinetic
+ information from such experiments. The cornerstone of our method
+ is a transformation of the rupture-force histograms obtained at
+ different force-loading rates into the force-dependent lifetimes
+ measurable in constant-force experiments. To interpret the
+ force-dependent lifetimes, we derive a generalization of Bell's
+ formula that is formally exact within the framework of Kramers
+ theory. This result complements the analytical expression for the
+ lifetime that we derived previously for a class of model
+ potentials. We illustrate our procedure by analyzing the nanopore
+ unzipping of DNA hairpins and the unfolding of a protein attached
+ by flexible linkers to an atomic force microscope. Our procedure
+ to transform rupture-force histograms into the force-dependent
+ lifetimes remains valid even when the molecular extension is a
+ poor reaction coordinate and higher-dimensional free-energy
+ surfaces must be considered. In this case the microscopic
+ interpretation of the lifetimes becomes more challenging because
+ the lifetimes can reveal richer, and even nonmonotonic, dependence
+ on the force.",
+ ISSN = "1091-6490",
+ doi = "10.1073/pnas.0806085105",
+ URL = "http://www.ncbi.nlm.nih.gov/pubmed/18852468",
+ language = "eng",
+}
+
@article { evans01,
author = EEvans,
title = "Probing the relation between force--lifetime--and chemistry in
issn = "1521-3773",
doi = "10.1002/1521-3773(20000915)39:18<3212::AID-ANIE3212>3.0.CO;2-X",
eprint = "",
- url = "http://dx.doi.org/10.1002/1521-3773(20000915)39:18<3212::AID-
- ANIE3212>3.0.CO;2-X",
+ url = "http://dx.doi.org/10.1002/1521-3773(20000915)39:18<3212::AID-ANIE3212>3.0.CO;2-X",
abstract = "How do molecules interact with each other? What happens if a
neurotransmitter binds to a ligand-operated ion channel? How do
antibodies recognize their antigens? Molecular recognition events play
number = 2,
pages = "159--166",
issn = "0141-8130",
- ISSN = "1879-0003",
+ alternative_issn = "1879-0003",
doi = "10.1016/j.ijbiomac.2009.12.001",
url = "http://www.sciencedirect.com/science/article/B6T7J-
4XWMND2-1/2/7ef768562b4157fc201d450553e5de5e",
potential of this new method for accurate and automated analysis of
force spectroscopy data and for novel automated screening techniques is
shown with bacteriorhodopsin and with protein constructs containing GFP
- and titin kinase."
+ and titin kinase.",
+ note = "Contour length space and barrier position fingerprinting.",
}
@article { raible04,
}
@article { hong10,
- author = XHong #" and "# and XChu #" and "# PZou #" and "# YLiu
+ author = XHong #" and "# XChu #" and "# PZou #" and "# YLiu
#" and "# GYang,
title = "Magnetic-field-assisted rapid ultrasensitive
immunoassays using Fe3{O4}/Zn{O}/Au nanorices as Raman
author = LiLi #" and "# BLi #" and "# GYang #" and "# CYLi,
title = "Polymer decoration on carbon nanotubes via physical
vapor deposition.",
- journal = LANG
+ journal = LANG,
year = 2007,
month = jul,
day = 31,
language = "eng",
}
-@article{ honda08
+@article{ honda08,
author = MHonda #" and "# YBaba #" and "# NHiaro #" and "# TSekiguchi,
title = "Metal-molecular interface of sulfur-containing amino acid
and thiophene on gold surface",
language = "eng",
}
-@article{ hager02
+@article{ hager02,
author = GHager #" and "# ABrolo,
title = "Adsorption/desorption behaviour of cysteine and cystine in
neutral and basic media: electrochemical evidence for differing
rate.",
note = "These guys seem to be pretty thorough, give this one another read.",
}
+
+@article{ baljon96,
+ author = ABaljon #" and "# MRobbins,
+ title = "Energy Dissipation During Rupture of Adhesive Bonds",
+ journal = SCI,
+ volume = 271,
+ number = 5248,
+ pages = "482--484",
+ year = 1996,
+ month = jan,
+ doi = "10.1126/science.271.5248.482",
+ URL = "http://www.sciencemag.org/content/271/5248/482.abstract",
+ eprint = "http://www.sciencemag.org/content/271/5248/482.full.pdf",
+ abstract = "Molecular dynamics simulations were used to study
+ energy-dissipation mechanisms during the rupture of a thin
+ adhesive bond formed by short chain molecules. The degree of
+ dissipation and its velocity dependence varied with the state of
+ the film. When the adhesive was in a liquid phase, dissipation was
+ caused by viscous loss. In glassy films, dissipation occurred
+ during a sequence of rapid structural rearrangements. Roughly
+ equal amounts of energy were dissipated in each of three types of
+ rapid motion: cavitation, plastic yield, and bridge rupture. These
+ mechanisms have similarities to nucleation, plastic flow, and
+ crazing in commercial polymeric adhesives.",
+}
+
+@article{ fisher99a,
+ author = TEFisher #" and "# PMarszalek #" and "# AOberhauser
+ #" and "# MCarrionVazquez #" and "# JFernandez,
+ title = "The micro-mechanics of single molecules studied with
+ atomic force microscopy.",
+ journal = JPhysio,
+ year = 1999,
+ month = oct,
+ day = 01,
+ address = "Department of Physiology and Biophysics, Mayo Foundation,
+ 1-117 Medical Sciences Building, Rochester, MN 55905, USA.",
+ volume = "520 Pt 1",
+ pages = "5--14",
+ keywords = "Animals",
+ keywords = "Extracellular Matrix",
+ keywords = "Extracellular Matrix Proteins",
+ keywords = "Humans",
+ keywords = "Microscopy, Atomic Force",
+ keywords = "Polysaccharides",
+ abstract = "The atomic force microscope (AFM) in its force-measuring
+ mode is capable of effecting displacements on an angstrom scale
+ (10 A = 1 nm) and measuring forces of a few piconewtons. Recent
+ experiments have applied AFM techniques to study the mechanical
+ properties of single biological polymers. These properties
+ contribute to the function of many proteins exposed to mechanical
+ strain, including components of the extracellular matrix
+ (ECM). The force-bearing proteins of the ECM typically contain
+ multiple tandem repeats of independently folded domains, a common
+ feature of proteins with structural and mechanical
+ roles. Polysaccharide moieties of adhesion glycoproteins such as
+ the selectins are also subject to strain. Force-induced extension
+ of both types of molecules with the AFM results in conformational
+ changes that could contribute to their mechanical function. The
+ force-extension curve for amylose exhibits a transition in
+ elasticity caused by the conversion of its glucopyranose rings
+ from the chair to the boat conformation. Extension of multi-domain
+ proteins causes sequential unraveling of domains, resulting in a
+ force-extension curve displaying a saw tooth pattern of peaks. The
+ engineering of multimeric proteins consisting of repeats of
+ identical domains has allowed detailed analysis of the mechanical
+ properties of single protein domains. Repetitive extension and
+ relaxation has enabled direct measurement of rates of domain
+ unfolding and refolding. The combination of site-directed
+ mutagenesis with AFM can be used to elucidate the amino acid
+ sequences that determine mechanical stability. The AFM thus offers
+ a novel way to explore the mechanical functions of proteins and
+ will be a useful tool for studying the micro-mechanics of
+ exocytosis.",
+ ISSN = "0022-3751",
+ URL = "http://www.ncbi.nlm.nih.gov/pubmed/10517795",
+ language = "eng",
+}
+
+@article{ fisher99b,
+ author = TEFisher #" and "# AOberhauser #" and "# MCarrionVazquez
+ #" and "# PMarszalek #" and "# JFernandez,
+ title = "The study of protein mechanics with the atomic force microscope.",
+ journal = "Trends in biochemical sciences",
+ year = "1999",
+ month = oct,
+ address = "Dept of Physiology and Biophysics, Mayo Foundation, 1-117
+ Medical Sciences Building, Rochester, MN 55905, USA.",
+ volume = 24,
+ number = 10,
+ pages = "379--384",
+ keywords = "Entropy",
+ keywords = "Kinetics",
+ keywords = "Microscopy, Atomic Force",
+ keywords = "Protein Binding",
+ keywords = "Protein Folding",
+ keywords = "Proteins",
+ abstract = "The unfolding and folding of single protein molecules
+ can be studied with an atomic force microscope (AFM). Many
+ proteins with mechanical functions contain multiple, individually
+ folded domains with similar structures. Protein engineering
+ techniques have enabled the construction and expression of
+ recombinant proteins that contain multiple copies of identical
+ domains. Thus, the AFM in combination with protein engineering
+ has enabled the kinetic analysis of the force-induced unfolding
+ and refolding of individual domains as well as the study of the
+ determinants of mechanical stability.",
+ ISSN = "0968-0004",
+ URL = "http://www.ncbi.nlm.nih.gov/pubmed/10500301",
+ language = "eng",
+}
+
+@article{ zlatanova00,
+ author = JZlatanova #" and "# SLindsay #" and "# SLeuba,
+ title = "Single molecule force spectroscopy in biology using the
+ atomic force microscope.",
+ journal = PBPMB,
+ year = 2000,
+ address = "Biochip Technology Center, Argonne National Laboratory,
+ 9700 South Cass Avenue, Bldg. 202-A253, Argonne, IL 60439,
+ USA. jzlatano@duke.poly.edu",
+ volume = 74,
+ number = "1--2",
+ pages = "37--61",
+ keywords = "Biophysics",
+ keywords = "Cell Adhesion",
+ keywords = "DNA",
+ keywords = "Elasticity",
+ keywords = "Microscopy, Atomic Force",
+ keywords = "Polysaccharides",
+ keywords = "Proteins",
+ keywords = "Signal Processing, Computer-Assisted",
+ keywords = "Viscosity",
+ abstract = "The importance of forces in biology has been recognized
+ for quite a while but only in the past decade have we acquired
+ instrumentation and methodology to directly measure interactive
+ forces at the level of single biological macromolecules and/or
+ their complexes. This review focuses on force measurements
+ performed with the atomic force microscope. A general introduction
+ to the principle of action is followed by review of the types of
+ interactions being studied, describing the main results and
+ discussing the biological implications.",
+ ISSN = "0079-6107",
+ URL = "http://www.ncbi.nlm.nih.gov/pubmed/11106806",
+ language = "eng",
+ note = "Lots of great force-clamp cartoons explaining different
+ approach/retract features.",
+}
+
+@article{ viani99,
+ author = MViani #" and "# TESchafer #" and "# AChand #" and "# MRief
+ #" and "# HEGaub #" and "# HHansma,
+ title = "Small cantilevers for force spectroscopy of single molecules",
+ journal = JAP,
+ year = 1999,
+ volume = 86,
+ number = 4,
+ pages = "2258--2262",
+ abstract = "We have used a simple process to fabricate small
+ rectangular cantilevers out of silicon nitride. They have lengths
+ of 9--50 $\mu$m, widths of 3--5 $\mu$m, and thicknesses of 86 and
+ 102 nm. We have added metallic reflector pads to some of the
+ cantilever ends to maximize reflectivity while minimizing
+ sensitivity to temperature changes. We have characterized small
+ cantilevers through their thermal spectra and show that they can
+ measure smaller forces than larger cantilevers with the same
+ spring constant because they have lower coefficients of viscous
+ damping. Finally, we show that small cantilevers can be used for
+ experiments requiring large measurement bandwidths, and have used
+ them to unfold single titin molecules over an order of magnitude
+ faster than previously reported with conventional cantilevers.",
+ ISSN = "0021-8979",
+ issn_online = "1089-7550",
+ doi = "10.1063/1.371039",
+ URL = "http://jap.aip.org/resource/1/japiau/v86/i4/p2258_s1",
+ language = "eng",
+}
+
+@article{ capitanio02,
+ author = MCapitanio #" and "# GRomano #" and "# RBallerini #" and "#
+ MGiuntini #" and "# FPavone #" and "# DDunlap #" and "# LFinzi,
+ title = "Calibration of optical tweezers with differential
+ interference contrast signals",
+ journal = RSI,
+ year = 2002,
+ volume = 73,
+ number = 4,
+ pages = "1687--1696",
+ abstract = "A comparison of different calibration methods for
+ optical tweezers with the differential interference contrast (DIC)
+ technique was performed to establish the uses and the advantages
+ of each method. A detailed experimental and theoretical analysis
+ of each method was performed with emphasis on the anisotropy
+ involved in the DIC technique and the noise components in the
+ detection. Finally, a time of flight method that permits the
+ reconstruction of the optical potential well was demonstrated.",
+ ISSN = "0034-6748",
+ issn_online = "1089-7623",
+ doi = "10.1063/1.1460929",
+ URL = "http://rsi.aip.org/resource/1/rsinak/v73/i4/p1687_s1",
+ language = "eng",
+}
+
+@article{ binnig86,
+ author = GBinnig #" and "# CQuate #" and "# CGerber,
+ title = "Atomic force microscope",
+ journal = PRL,
+ year = 1986,
+ month = mar,
+ day = 03,
+ volume = 56,
+ number = 9,
+ pages = "930--933",
+ abstract = "The scanning tunneling microscope is proposed as a
+ method to measure forces as small as $10^{-18}$ N. As one
+ application for this concept, we introduce a new type of
+ microscope capable of investigating surfaces of insulators on an
+ atomic scale. The atomic force microscope is a combination of the
+ principles of the scanning tunneling microscope and the stylus
+ profilometer. It incorporates a probe that does not damage the
+ surface. Our preliminary results in air demonstrate a lateral
+ resolution of 30 \AA and a vertical resolution less than 1 \AA.",
+ ISSN = "1079-7114",
+ doi = "10.1103/PhysRevLett.56.930",
+ URL = "http://www.ncbi.nlm.nih.gov/pubmed/10033323",
+ language = "eng",
+ note = "Original AFM paper.",
+}
+
+@article{ williams86,
+ author = CCWilliams #" and "# HWickramasinghe,
+ title = "Scanning thermal profiler",
+ journal = APL,
+ year = 1986,
+ month = dec,
+ day = 8,
+ volume = 49,
+ number = 23,
+ pages = "1587--1589",
+ abstract = "A new high‐resolution profilometer has been demonstrated
+ based upon a noncontacting near‐field thermal probe. The thermal
+ probe consists of a thermocouple sensor with dimensions
+ approaching 100 nm. Profiling is achieved by scanning the heated
+ sensor above but close to the surface of a solid. The conduction
+ of heat between tip and sample via the air provides a means for
+ maintaining the sample spacing constant during the lateral
+ scan. The large difference in thermal properties between air and
+ solids makes the profiling technique essentially independent of
+ the material properties of the solid. Noncontact profiling of
+ resist and metal films has shown a lateral resolution of 100 nm
+ and a depth solution of 3 nm. The basic theory of the new probe is
+ described and the results presented.",
+ issn = "0003-6951",
+ issn_online = "1077-3118",
+ doi = "10.1063/1.97288",
+ URL = "http://apl.aip.org/resource/1/applab/v49/i23/p1587_s1",
+ language = "eng",
+}
+
+@article{ martin87,
+ author = MMartin #" and "# CCWilliams #" and "# HWickramasinghe,
+ title = "Atomic force microscope-force mapping and profiling on a sub
+ 100-\AA scale",
+ journal = JAP,
+ year = 1987,
+ month = may,
+ day = 15,
+ volume = 61,
+ number = 10,
+ pages = "4723--4729",
+ abstract = "A modified version of the atomic force microscope is
+ introduced that enables a precise measurement of the force between
+ a tip and a sample over a tip‐sample distance range of 30--150
+ \AA. As an application, the force signal is used to maintain the
+ tip‐sample spacing constant, so that profiling can be achieved
+ with a spatial resolution of 50 \AA. A second scheme allows the
+ simultaneous measurement of force and surface profile; this scheme
+ has been used to obtain material-dependent information from
+ surfaces of electronic materials.",
+ issn = "0021-8979",
+ issn_online = "1089-7550",
+ doi = "10.1063/1.338807"
+ URL = "http://jap.aip.org/resource/1/japiau/v61/i10/p4723_s1",
+ language = "eng",
+}
+
+@article{ meyer88
+ author = GMeyer #" and "# NMAmer,
+ title = "Novel optical approach to atomic force microscopy",
+ journal = APL,
+ year = 1988,
+ month = sep,
+ day = 19,
+ volume = 53,
+ number = 12,
+ pages = "1045--1047",
+ abstract = "A sensitive and simple optical method for detecting the
+ cantilever deflection in atomic force microscopy is described. The
+ method was incorporated in an atomic force microscope, and imaging
+ and force measurements, in ultrahigh vacuum, were successfully
+ performed.",
+ issn = "0003-6951",
+ issn_online = "1077-3118",
+ doi = "10.1063/1.100061",
+ URL = "http://apl.aip.org/resource/1/applab/v53/i12/p1045_s1",
+ language = "eng",
+}
\include{blurb/abstract}
\end{preamble}
-\linenumbers
+\iffinal{}{\linenumbers}
\begin{thesis}
\pdfbookmark[-1]{Mainmatter}{Mainmatter}
\include{introduction/main}
\include{apparatus/main}
-\include{unfolding/main}
\include{sawsim/main}
\include{temperature/main}
\include{cantilever/main}
\bibliography{%
apparatus/main,%
-% unfolding-distributions/main,
% sawsim/main,% currently empty
cantilever-calib/main,%
packaging/main,%
\label{sec:sawsim:results}
\subsection{Force curves generated by simulation}
-\label{sec:sawsim:results-force_curves}
+\label{sec:sawsim:results:force-curves}
\Cref{fig:sawsim:sim-sawtooth} shows three simulated force curves from
pulling a polymer composed of eight identical protein molecules using
(\cref{fig:expt-sawtooth}), the forces at which identical protein
molecules unfold fluctuate, revealing the stochastic nature of protein
unfolding since no instrumental noise is included in the simulation.
-\Cref{fig:sawsim:sim-hist} shows the distribution of the unfolding forces,
-\ie, the highest force in each peak (except the last peak in a force
-curve), from a total of $400$ force curves ($3200$ force values). The
-unfolding forces have an average of $281\U{pN}$ with a standard
-deviation of $25\U{pN}$.
-% DONE: discuss noise? no.
+
+After aquiring a series of experimental unfolding curves, we need to
+fit the data to an explanatory model. For velocity-clamp experiments
+(\cref{sec:procedure,sec:sawsim:velocity-clamp}), we extract unfolding
+forces from the sawtooth curves (\cref{sec:hooke}) and generate
+histograms of unfolding forces. Then we construct a parameterized
+model of the experimental system (\cref{tab:sawsim:model}). We can
+then run \insilico\ experiments mimicking our \invitro\ experiments
+(\cref{fig:sawsim:sim-hist}). We extract the model parameters which
+provide the best fit using a ``fit quality'' metric and a nonlinear
+optimization routine (or a full parameter space sweep, for
+low-dimensional parameter spaces).
+
+\begin{table}[btp]
+ \begin{center}
+ \subfloat[][]{\label{tab:sawsim:domains}
+ \begin{tabular}{l l l l}
+ \toprule
+ \multicolumn{4}{c}{Domain states} \\
+ \midrule
+ Domain name & Initial count & Tension model & Model parameters \\
+ \midrule
+ AFM cantilever & 1 & Hooke (\cref{eq:sawsim:hooke}) &
+ $k_c=0.05\U{N/m}$ \\
+ Folded I27 & 8 & WLC (\cref{eq:sawsim:wlc}) &
+ $p=3.9\U{\AA}$, $L=5.1\U{nm}$ \\
+ Unfolded I27 & 0 & WLC (\cref{eq:sawsim:wlc}) &
+ $p=3.9\U{\AA}$, $L=33.8\U{nm}$ \\
+ \bottomrule
+ \end{tabular}} \\
+ \subfloat[][]{\label{tab:sawsim:transitions}
+ \begin{tabular}{l l l l l}
+ \toprule
+ \multicolumn{5}{c}{Transition rates} \\
+ \midrule
+ Transition & Source & Target & Rate model & Model parameters \\
+ \midrule
+ Unfolding & Folded I27 & Unfolded I27 & Bell (\cref{eq:sawsim:bell}) &
+ $k_{u0}=3.3\E{-4}\U{s$^{-1}$}$, $\Delta x=0.35\U{nm}$. \\
+ \bottomrule
+ \end{tabular}}
+ \caption{\subref{tab:sawsim:domains} Model for
+ I27\textsubscript{8} domain states and
+ \subref{tab:sawsim:transitions} transitions between them
+ (compare with \cref{fig:sawsim:domains}). The models and
+ parameters are those given by \citet{carrion-vazquez99b}.
+ \citet{carrion-vazquez99b} don't list their cantilever spring
+ constant (or, presumably, use it in their simulations), but we
+ can estimate it from the rebound slope in their Figures~2.a and
+ 2.b, see
+ \cref{fig:sawsim:kappa-sawteeth}.\label{tab:sawsim:model}}
+ \end{center}
+\end{table}
+
+Because the unfolding behavious of an individual sawtooth curve is
+stochastic (\cref{fig:sawsim:sim-sawtooth}), we cannot directly
+compare single curves in our fit quality metric. Instead, we gather
+many experimental and simulated curves, and compare the aggregate
+properties. For velocity-clamp experiments, the usual aggregate
+property used for comparison is a histogram of unfolding
+forces\citep{carrion-vazquez99b} (\cref{fig:sawsim:sim-hist}).
+Defining and extracting “unfolding force” is suprisingly complicated
+(\cref{sec:hooke:unfolding-force}), but basically it is the highest
+tension force achieved by the chain before an unfolding event (the
+drops in the sawtooth). The final drop is not an unfolding event, it
+is the entire chain breaking away from the cantilever tip, severing
+the connection between the substrate and the cantilever.
\begin{figure}
-\vspace{-1in}
-\begin{center}
-\subfloat[][]{\asyinclude{figures/sim-sawtooth/sim-sawtooth}%
- \label{fig:sawsim:sim-sawtooth}%
-}\\
-\subfloat[][]{\asyinclude{figures/sim-hist/sim-hist}%
- \label{fig:sawsim:sim-hist}%
-}
-\caption{(a) Three simulated force curves from pulling a polymer of
- eight identical protein molecules. The simulation was carried out
- using the parameters: pulling speed $v=1\U{$\mu$m/s}$, cantilever
- spring constant $\kappa_c=50\U{pN/nm}$, temperature $T=300\U{K}$,
- persistence length of unfolded proteins $p_u=0.40\U{nm}$, $\Delta
- x_u=0.225\U{nm}$, and $k_{u0}=5\E{-5}\U{s$^{-1}$}$. The contour
- length between the two linking points on a protein molecule is
- $L_{f1}=3.7\U{nm}$ in the folded form and $L_{u1}=28.1\U{nm}$ in the
- unfolded form. These parameters are those of ubiquitin molecules
- connected through the N-C termini\citep{chyan04,carrion-vazquez03}.
- Detachment from the tip or substrate is assumed to occur at a force
- of $400\U{pN}$. In experiments, detachments have been observed to
- occur at a variety of forces. For clarity, the green and blue
- curves are offset by $200$ and $400\U{pN}$ respectively. (b) The
- distribution of the unfolding forces from $400$ simulated force
- curves ($3200$ data points) such as that shown in (a). The frequency
- is normalized by the total number of points, \ie, the height of each
- bin is equal to the number of data points in that bin divided by the
- total number of data points ($3200$, for this
- histogram).\label{fig:sawsim:sim-all}}
-\end{center}
+ \begin{center}
+ \subfloat[][]{\asyinclude{figures/sim-sawtooth/sim-sawtooth}%
+ \label{fig:sawsim:sim-sawtooth}%
+ }
+ \hspace{.25in}%
+ \subfloat[][]{\asyinclude{figures/sim-hist/sim-hist}%
+ \label{fig:sawsim:sim-hist}%
+ }
+ \caption{\subref{fig:sawsim:sim-sawtooth} Three simulated force
+ curves from pulling a polymer of eight identical protein
+ molecules. The simulation was carried out using the parameters:
+ pulling speed $v=1\U{$\mu$m/s}$, cantilever spring constant
+ $\kappa_c=50\U{pN/nm}$, temperature $T=300\U{K}$, persistence
+ length of unfolded proteins $p_u=0.40\U{nm}$, $\Delta
+ x_u=0.225\U{nm}$, and $k_{u0}=5\E{-5}\U{s$^{-1}$}$. The contour
+ length between the two linking points on a protein molecule is
+ $L_{f1}=3.7\U{nm}$ in the folded form and $L_{u1}=28.1\U{nm}$ in
+ the unfolded form. These parameters are those of ubiquitin
+ molecules connected through the N-C
+ termini\citep{chyan04,carrion-vazquez03}. Detachment from the
+ tip or substrate is assumed to occur at a force of $400\U{pN}$.
+ In experiments, detachments have been observed to occur at a
+ variety of forces. For clarity, the green and blue curves are
+ offset by $200$ and $400\U{pN}$ respectively.
+ \subref{fig:sawsim:sim-hist} The distribution of the unfolding
+ forces from $400$ simulated force curves ($3200$ data points)
+ such as those shown in \subref{fig:sawsim:sim-sawtooth}. The
+ frequency is normalized by the total number of points, \ie, the
+ height of each bin is equal to the number of data points in that
+ bin divided by the total number of data
+ points.\label{fig:sawsim:sim-all}}
+ \end{center}
\end{figure}
-\subsection{Dependence of the unfolding force on the unfolding order
-and polymer length}
-\label{sec:sawsim:results-scaffold}
+\subsection{The supramolecular scaffold}
+\label{sec:sawsim:results:scaffold}
Analysis of the mechanical unfolding data is complicated by the
dependence of the average unfolding force on the unfolding order due
to the serial linkage of the molecules. Under an external stretching
force $F$, the probability of some domain unfolding in a polymer with
-$N_f$ folded domains is $N_fP_1$ (\cref{eq:sawsim:prob-n}), which is higher
-than the unfolding probability for a single molecule $P_1$.
+$N_f$ folded domains is $N_fP_1$ (\cref{eq:sawsim:prob-n}), which is
+higher than the unfolding probability for a single molecule $P_1$.
Consequently, the average unfolding force is lower for the earlier
unfolding events when $N_f$ is larger, and the force should increase
as more and more molecules become unfolded. However, there is a
competing factor that opposes this trend. As the protein molecules
unfold, the chain becomes softer and the force loading rate becomes
-lower when the pulling speed is constant, leading to a decrease in the
-unfolding force. The dependence of the average unfolding force on the
-unfolding order is the result of these two opposing effects.
-\Cref{fig:sawsim:order-dep} shows the dependence of the average unfolding
-force on the unfolding force peak order (the temporal order of
-unfolding events) for four polymers with $4$, $8$, $12$, and $16$
-identical protein molecules. The effect of polymer chain softening
-dominates the initial unfolding events, and the average unfolding
-force decreases as more molecules unfold. After several molecules
-have unfolded, the softening for each additional unfolding event
-becomes less significant, the change in unfolding probability becomes
-dominant, and the unfolding force increases upon each subsequent
-unfolding event\citep{zinober02}.
+lower when the pulling speed is constant. This lower loading rate
+leads to a decrease in the unfolding force (in the no-loading limit,
+all unfolding events occur at a tension of $0\U{N}$). The dependence
+of the average unfolding force on the unfolding order is the result of
+these two opposing effects. \Cref{fig:sawsim:order-dep} shows the
+dependence of the average unfolding force on the unfolding force peak
+order (the temporal order of unfolding events) for four polymers with
+$4$, $8$, $12$, and $16$ identical protein molecules. The effect of
+polymer chain softening dominates the initial unfolding events, and
+the average unfolding force decreases as more molecules unfold. After
+several molecules have unfolded, the softening for each additional
+unfolding event becomes less significant, the change in unfolding
+probability becomes dominant, and the unfolding force increases upon
+each subsequent unfolding event\citep{zinober02}.
We validate this explanation by calculating the unfolding force
-probability distribution's depending on the two competing factors.
+probability distribution's dependence on the two competing factors.
The rate of unfolding events with respect to force is
\begin{align}
r_{uF} &= -\deriv{F}{N_f}
effective spring constant of one unfolded domain, assumed constant for
a particular polymer/cantilever combination), $\kappa v$ is the force
loading rate, and $k_u$ is the unfolding rate constant
-(\cref{eq:sawsim:bell}). In the last expression, $\rho\equiv k_BT/\Delta
-x_u$, and $\alpha\equiv-\rho\ln(N_fk_{u0}\rho/\kappa v)$. The event
-probability density for events with an exponentially increasing
-likelihood function follows the Gumbel (minimum) probability
-density\citep{NIST:gumbel}, with $\rho$ and $\alpha$ being the scale
-and location parameters, respectively
+(\cref{eq:sawsim:bell}). In the last expression, $\rho\equiv
+k_BT/\Delta x_u$, and $\alpha\equiv-\rho\ln(N_fk_{u0}\rho/\kappa v)$.
+The event probability density for events with an exponentially
+increasing likelihood function follows the Gumbel (minimum)
+probability density\citep{NIST:gumbel} with $\rho$ and $\alpha$ being
+the scale and location parameters, respectively\citep{hummer03}
\begin{equation}
\mathcal{P}(F) = \frac{1}{\rho} \exp\p[{\frac{F-\alpha}{\rho}
-\exp\p({\frac{F-\alpha}{\rho}})
}] \;. \label{eq:sawsim:gumbel}
\end{equation}
The distribution has a mean $\avg{F}=\alpha-\gamma_e\rho$ and a
-variance $\sigma^2=\pi^2\rho^2/6$, where $\gamma_e=0.577\ldots$ is the
-Euler-Mascheroni constant. Therefore, the unfolding force
-distribution has a variance $\sigma^2 = (\pi k_BT/\Delta x_u)^2/6$,
-and and average
+variance $\sigma^2 = \pi^2\rho^2/6$, where $\gamma_e=0.577\ldots$ is
+the Euler-Mascheroni constant. Therefore, the unfolding force
+distribution has a variance
+\begin{equation}
+ \sigma^2 = \frac{\p({\frac{\pi k_BT}{\Delta x_u}})^2}{6} \;,
+ \label{eq:sawsim:variance}
+\end{equation}
+and and average\citep{brockwell02,hummer03}
+% TODO: verify brockwell equivalence (p465)
\begin{equation}
\avg{F(i)} = \frac{k_BT}{\Delta x_u}
- \p[{-\ln\p({\frac{N_fk_{u0}k_BT}{\kappa v\Delta x_u}})
+ \p[{\ln\p({\frac{\kappa v\Delta x_u}{N_fk_{u0}k_BT}})
-\gamma_e}] \;, \label{eq:sawsim:order-dep}
\end{equation}
-% This is discussed in brockwell02, p465
-% consolidate with src/unfolding/distributions-single_domain-constant_loading.tex
-where $N_f$ and $\kappa$ depend on the domain index $i=N_u$. Curves based
-on this formula fit the simulated data remarkably well considering the
-effective WLC\index{WLC} stiffness $\kappa_\text{WLC}$ is the only fitted
-parameter, and that the actual WLC stiffness is not constant, as we
-have assumed here, but a non-linear function of $F$.
-
-From \cref{fig:sawsim:order-dep}, it can be seen that the proper way to
+where $N_f$ and $\kappa$ depend on the domain index $i=N_u$. Curves
+based on this formula fit the simulated data remarkably well
+considering the effective WLC\index{WLC} stiffness $\kappa_\text{WLC}$
+is the only fitted parameter, and that the actual WLC stiffness is not
+constant, as we have assumed here, but a non-linear function of $F$.
+\citet{dudko08} derived a formula for the loading rate for a WLC, but
+as far as I know, nobody has found an analytical form for the
+unfolding force histograms produced under such a variable loading
+rate.
+\nomenclature{$r_{uF}$}{Unfolding loading rate (newtons per second)}
+\nomenclature{$\gamma_e$}{Euler-Macheroni constant, $\gamma_e=0.577\ldots$}
+
+From \cref{fig:sawsim:order-dep}, we see that the proper way to
process data from mechanical unfolding experiments is to group the
curves according to the length of the polymer and to perform
statistical analysis separately for peaks with the same unfolding
order. However, in most experiments, the tethering of the polymer to
the AFM tip is by nonspecific adsorption; as a result, the polymers
being stretched between the tip and the substrate have various
-lengths. In addition, the interactions between the tip and the
-surface often cause irregular features in the beginning of the force
-curve (\cref{fig:expt-sawtooth}), making the identification of the
-first peak uncertain. Furthermore, it is often difficult to acquire a
-large amount of data in single molecule experiments. These
-difficulties make the aforementioned data analysis approach unfeasible
-for many mechanical unfolding experiments. As a result, the values of
-all force peaks from polymers of different lengths are often pooled
-together for statistical analysis. To assess the errors caused by
-such pooling, simulation data were analyzed using different pooling
-methods and the results were compared. \Cref{fig:sawsim:sim-hist} shows
-that, for a polymer with eight protein molecules, the average
-unfolding force is $281\U{pN}$ with a standard deviation of $25\U{pN}$
-when all data is pooled. If only the first peaks in the force curves
-are analyzed, the average force is $279\U{pN}$ with a standard
-deviation of $22\U{pN}$. While for the fourth and eighth peaks, the
-average force are $275\U{pN}$ and $300\U{pN}$, respectively, and the
-standard deviations are $23\U{pN}$ and $25\U{pN}$, respectively. As
-expected from the Gumbel distribution, the width of the unfolding
-force distribution (insets in \cref{fig:sawsim:order-dep}) is only weakly
-effected by unfolding order, but the average unfolding force can be
-quite different for the same protein because of the differences in
-unfolding order and polymer length.
+lengths\citep{1li00}. In addition, the interactions between the tip
+and the surface often cause irregular features in the beginning of the
+force curve (\cref{fig:expt-sawtooth}), making the identification of
+the first peak uncertain\citep{carrion-vazquez00}. Furthermore, it is
+often difficult to acquire a large amount of data in single molecule
+experiments. These difficulties make the aforementioned data analysis
+approach unfeasible for many mechanical unfolding experiments. As a
+result, the values of all force peaks from polymers of different
+lengths are often pooled together for statistical analysis. To assess
+the errors caused by such pooling, simulation data were analyzed using
+different pooling methods and the results were compared.
+\Cref{fig:sawsim:sim-hist} shows that, for a polymer with eight
+protein molecules, the average unfolding force is $281\U{pN}$ with a
+standard deviation of $25\U{pN}$ when all data is pooled. If only the
+first peaks in the force curves are analyzed, the average force is
+$279\U{pN}$ with a standard deviation of $22\U{pN}$. While for the
+fourth and eighth peaks, the average force are $275\U{pN}$ and
+$300\U{pN}$, respectively, and the standard deviations are $23\U{pN}$
+and $25\U{pN}$, respectively. As expected from the Gumbel
+distribution, the width of the unfolding force distribution (insets in
+\cref{fig:sawsim:order-dep}) is only weakly effected by unfolding
+order, but the average unfolding force can be quite different for the
+same protein because of the differences in unfolding order and polymer
+length.
\begin{figure}
\begin{center}
force constants are available. In addition, different single molecule
manipulation techniques, such as the AFM and laser tweezers, differ
mainly in the range of the spring constants of their force
-transducers. \Cref{fig:sawsim:kappa-sawteeth} shows the simulated force
-curves from pulling an octamer of protein molecules using cantilevers
-with different force constants, while other parameters are identical.
-For this model protein, the appearance of the force curve does not
-change much until the force constant of the cantilever reaches a
-certain value ($\kappa_c=50\U{pN/nm}$). When $\kappa_c$ is lower than
-this value, the individual unfolding events become less identifiable.
-In order to observe individual unfolding events, the cantilever needs
-to have a force constant high enough so that the bending at the
-maximum force is small in comparison with the contour length increment
-from the unfolding of a single molecule. \Cref{fig:sawsim:kappa-sawteeth}
-also shows that the back side of the force peaks becomes more tilted
-as the cantilever becomes softer. This is due to the fact that the
-extension (end-to-end distance) of the protein polymer has a large
-sudden increase as the tension rebalances after an unfolding event.
+transducers\citep{walton08}. \Cref{fig:sawsim:kappa-sawteeth} shows
+the simulated force curves from pulling an octamer of protein
+molecules using cantilevers with different force constants, while
+other parameters are identical. For this model protein, the
+appearance of the force curve does not change much until the force
+constant of the cantilever reaches a certain value
+($\kappa_c=50\U{pN/nm}$). When $\kappa_c$ is lower than this value,
+the individual unfolding events become less identifiable. In order to
+observe individual unfolding events, the cantilever needs to have a
+force constant high enough so that the bending at the maximum force is
+small in comparison with the contour length increment from the
+unfolding of a single molecule. \Cref{fig:sawsim:kappa-sawteeth} also
+shows that the back side of the force peaks becomes more tilted as the
+cantilever becomes softer. This is due to the fact that the extension
+(end-to-end distance) of the protein polymer has a large sudden
+increase as the tension rebalances after an unfolding event.
It should also be mentioned that the contour length increment from
each unfolding event is not equal to the distance between adjacent
\end{figure}
\subsection{Determination of $\Delta x_u$ and $k_{u0}$}
-\label{sec:sawsim:results-fitting}
-
-The zero-force unfolding rate $k_{u0}$ and the distance $\Delta x_u$
-from the native state to the transition state are the two kinetic
-parameters obtainable for mechanical unfolding experiments by matching
-the simulated data with measured results. \cref{fig:sawsim:v-dep} shows the
-dependence of the unfolding force on the pulling speed for different
-values of $k_{u0}$ and $\Delta x_u$. As expected, the unfolding force
-increases linearly with the pulling speed in the linear-log
-plot\citep{evans99}. While the magnitude of the unfolding forces is
-affected by both $k_{u0}$ and $\Delta x_u$, the slope of speed
-dependence is primarily determined by $\Delta x_u$.
-\Cref{fig:sawsim:width-v-dep} shows that the width of the unfolding force
-distribution is very sensitive to $\Delta x_u$, as expected from the
-Gumbel distribution discussed in \cref{sec:sawsim:results-scaffold}. To
-obtain the values of $k_{u0}$ and $\Delta x_u$ for the protein, the
-pulling speed dependence and the distribution of the unfolding forces
-from simulation, such as those shown in \cref{fig:sawsim:v-dep} and the
-insets of \cref{fig:sawsim:width-v-dep}, are compared with the experimentally
+\label{sec:sawsim:results:fitting}
+
+As mentioned in \cref{sec:sawsim:results:force-curves}, fitting
+experimental unfolding force histograms to simulated histograms allows
+you to extract best-fit parameters for your simulation model. For
+example, if you have Bell model unfolding
+(\cref{sec:sawsim:rate:bell}), your two fitting parameters are the
+zero-force unfolding rate $k_{u0}$ and the distance $\Delta x_u$ from
+the native state to the transition state. \cref{fig:sawsim:v-dep}
+shows the dependence of the unfolding force on the pulling speed for
+different values of $k_{u0}$ and $\Delta x_u$. As expected, the
+unfolding force increases linearly with the pulling speed in the
+linear-log plot\citep{evans99}. While the magnitude of the unfolding
+forces is affected by both $k_{u0}$ and $\Delta x_u$, the slope of
+speed dependence is primarily determined by $\Delta x_u$
+(\cref{eq:sawsim:order-dep}). \Cref{fig:sawsim:width-v-dep} shows
+that the width of the unfolding force distribution is very sensitive
+to $\Delta x_u$, as expected from the Gumbel distribution
+(\cref{eq:sawsim:variance}). To obtain the values of $k_{u0}$ and
+$\Delta x_u$ for the protein, the pulling speed dependence and the
+distribution of the unfolding forces from simulation, such as those
+shown in \cref{fig:sawsim:v-dep} and the insets of
+\cref{fig:sawsim:width-v-dep}, are compared with the experimentally
measured results. The values of $k_{u0}$ and $\Delta x_u$ that
provide the best match are designated as the parameters describing the
protein under study. Since $k_{u0}$ and $\Delta x_u$ affect the
unfolding forces differently, the values of both parameters can be
determined simultaneously. The data used in plotting
-\cref{fig:sawsim:all-v-dep} includes all force peaks from the simulated force
-curves because most experimental data is analyzed that way.
-
-In most published literature, determination of the values of $k_{u0}$
-and $\Delta x_u$ was mostly done by carrying out simulations using a
-handful of possible unfolding parameters and selected the best fit by
-eye%
-%\citep{us,CV1999}
+\cref{fig:sawsim:all-v-dep} includes all force peaks from the
+simulated force curves because most experimental data is analyzed that
+way. % TODO: all? most data analyzed what way?
+
+In most published literature, $k_{u0}$ and $\Delta x_u$ were fit by
+carrying out simulations using a handful of possible unfolding
+parameters and selected the best fit by eye%
+%\citep{us,carrion-vazques99b,schlierf06}
. This approach does not allow estimation of uncertainties in the
fitting parameters, as shown by \citet{best02}. A more rigorous
approach involves quantifying the quality of fit between the
experimental and simulated force distributions, allowing the use of a
numerical minimization algorithm to pick the best fit parameters. We
-use the Jensen-Shannon divergence\citep{sims09,lin91}, a measure of
+use the Jensen--Shannon divergence\citep{sims09,lin91}, a measure of
the similarity between two probability distributions.
\begin{equation}
D_\text{JS}(p_e, p_s)
% simply sum residuals computed for each velocity, although it would
% also be reasonable to weight this sum according to the number of
% experimental unfolding events recorded for each velocity.
-% DONE: mention $D_\text{JS}$ features to explains selection over $\chi^2$? no.
-\Cref{fig:sawsim:fit-space} shows the Jensen-Shannon divergence calculated
-using \cref{eq:sawsim:D_JS} between an experimental data set and simulation
-results obtained using a range of values of $k_{u0}$ and $\Delta x_u$.
-There is an order of magnitude range of $k_{u0}$ that produce
-reasonable fits to experimental data (\cref{fig:sawsim:fit-space}), which is
-consistent with the results \citet{best02} obtained using a chi-square
-test. The values of $k_{u0}$ and $\Delta x_u$ can be determined to
-higher precision by using both the pulling speed dependent data and
-the unfolding force distribution, as well as any relevant information
-about the protein from other sources.
+
+The major advantage of the Jensen--Shannon divergence is that
+$D_\text{JS}$ is bounded ($0\le D_\text{JS}\le 1$) regardless of the
+experimental and simulated histograms. For comparison, Pearson's
+$\chi^2$ test,
+\begin{equation}
+ D_{χ^2} = \sum_i \frac{(p_e(i)-p_s(i))^2}{p_s(i)}) \;, \label{eq:sawsim:X2}
+\end{equation}
+is infinite if there is a bin for which $p_e(i)>0$ but $p_s(i)=0$.
+
+\Cref{fig:sawsim:fit-space} shows the Jensen--Shannon divergence
+calculated using \cref{eq:sawsim:D_JS} between an experimental data
+set and simulation results obtained using a range of values of
+$k_{u0}$ and $\Delta x_u$. There is an order of magnitude range of
+$k_{u0}$ that produce reasonable fits to experimental data
+(\cref{fig:sawsim:fit-space}), which is consistent with the results
+\citet{best02} obtained using a chi-square test. The values of
+$k_{u0}$ and $\Delta x_u$ can be determined to higher precision by
+using both the pulling speed dependent data and the unfolding force
+distribution, as well as any relevant information about the protein
+from other sources.
\begin{figure}
\begin{center}
\subfloat[][]{\asyinclude{figures/v-dep/v-dep-sd}%
\label{fig:sawsim:width-v-dep}%
}
- \caption{(a) The dependence of the unfolding forces on the pulling
- speed for three different model protein molecules characterized by
- the parameters $k_{u0}$ and $\Delta x_u$. The polymer length is
- eight molecules, and each symbol is the average of $3200$ data
- points. (b) The dependence of standard deviation of the unfolding
- force distribution on the pulling speed for the simulation data
- shown in (a), using the same symbols. The insets show the force
- distribution histograms for the three proteins at the pulling
- speed of $1\U{$\mu$m/s}$. The left, middle and right histograms
- are for the proteins represented by the top, middle, and bottom
- lines in (a), respectively.\label{fig:sawsim:all-v-dep}}
+ \caption{\subref{fig:sawsim:v-dep} The dependence of the unfolding
+ forces on the pulling speed for three different model protein
+ molecules characterized by the parameters $k_{u0}$ and $\Delta
+ x_u$. The polymer length is eight molecules, and each symbol is
+ the average of $3200$ data points.
+ \subref{fig:sawsim:width-v-dep} The dependence of standard
+ deviation of the unfolding force distribution on the pulling speed
+ for the simulation data shown in \subref{fig:sawsim:v-dep}, using
+ the same symbols. The insets show the force distribution
+ histograms for the three proteins at the pulling speed of
+ $1\U{$\mu$m/s}$. The left, middle and right histograms are for
+ the proteins represented by the top, middle, and bottom lines in
+ \subref{fig:sawsim:v-dep},
+ respectively.\label{fig:sawsim:all-v-dep}}
\end{center}
\end{figure}
parameter pair.\label{fig:sawsim:fit-space}}
\end{center}
\end{figure}
+
+\subsection{Features}
+\label{sec:sawsim:features}
+
+\sawsim\ is a great improvement over existing work in this field.
+\citet{best02} are the only authors to mention such automatic
+simulation comparisons, and their $\chi^2$ fit only compares mean
+unfolding forces over a range of speeds. They calculate $\avg{F}$
+through an iterative method, and assume a standard deviation of
+$20\U{pN}$ on their simulated $\avg{F}$. \sawsim, by comparison,
+makes full use of your experimental histograms, which you specify in a
+plain-text histogram file:
+\begin{center}
+\begin{spacing}{1}
+\begin{Verbatim}
+#HISTOGRAM: -v 6e-7
+#Force (N) Unfolding events
+1.4e-10 1
+1.5e-10 0
+\ldots
+3e-10 116
+3.1e-10 18
+3.2e-10 1
+
+#HISTOGRAM: -v 8e-7
+#Force (N) Unfolding events
+1.4e-10 0
+1.5e-10 3
+\ldots
+3.2e-10 50
+3.3e-10 13
+
+#HISTOGRAM: -v 1e-6
+#Force (N) Unfolding events
+1.5e-10 2
+1.6e-10 3
+\ldots
+3.3e-10 24
+3.4e-10 2
+\end{Verbatim}
+\end{spacing}
+\end{center}
+
+Each \sawsim\ run simulates a single sawtooth curve, so you need to
+run many \sawsim\ instances to generate your simulated histograms. To
+automate this task, \sawsim\ comes with a \citet{Python} wrapping
+library (\pysawsim), which provides convenient programmatic and
+command line interfaces for generating and manipulating \sawsim\ runs.
+For example, to compare the experimental histograms listed above with
+simulated data over a 50-by-50 grid of $k_{u0}$ and $\Delta x$, you
+would use something like
+\begin{Verbatim}[samepage]
+$ sawsim_hist_scan.py -f '-s cantilever,hooke,0.05 -N1 -s folded,null -N8
+ -s "unfolded,wlc,{0.39e-9,28e-9}" -k "folded,unfolded,bell,{%g,x%g}"
+ -q folded' -r '[1e-5,1e-3,50],[0.1e-9,1e-9,50]' --logx histograms.txt
+\end{Verbatim}
+That's a bit of a mouthful, so let's break it down. Without the
+\sawsim\ template (\Verb+-f ...+), we can focus on the comparison
+options:
+\begin{Verbatim}[samepage]
+$ sawsim_hist_scan.py \ldots -r '[1e-5,1e-3,50],[0.1e-9,1e-9,50]' --logx histograms.txt
+\end{Verbatim}
+This sets up a two-parameter sweep, with the first parameter going
+from $1\E{-5}$ to $1\E{-3}$ in 50 logarithmic steps, and the second
+going from $0.1\E{-9}$ to $1\E{-9}$ in 50 linear steps. The
+\sawsim\ template defines the simulation model
+(\cref{fig:sawsim:domains,tab:sawsim:model}), and \Verb+%g+ marks the
+location where the swept parameters will be inserted.
+
+Behind the scenes, \pysawsim\ is spawning several concurrent
+\sawsim\ processes to take advantage of any parallel processing
+facilities you may have access to (e.g.~multiple cores, MPI, PBS,
+\ldots). A 50-by-50 grid with 400 runs per pixel at about one second
+per \sawsim\ pull would take arount 12 days of serial execution.
+Moving the simulation to the departments' 16 core file server cuts
+that execution time down to 18 hours, which will easily complete over
+a quiet weekend. Using MPI on the departments' 15 box, dual core
+computer lab, the simulation would finish overnight.
+\nomenclature{MPI}{Message passing interface, a parallel computing
+ infrastructure}
+\nomenclature{PBS}{Portable batch system, a parallel computing
+ infrastructure. You should be able to distinguish this from the
+ other PBS (phosphate buffered saline) based on the context}
+
+\sawsim\ also takes advantage of a number of optimizations for faster
+execution. One of the bottlenecks in the \sawsim\ code is the TODO:
+interpolating tree.
+
+\subsection{Testing}
+\label{sec:sawsim:testing}
+
+Once a body of code reaches a certain level of complication, it
+becomes difficult to convince others (or yourself) that it's actually
+working correctly. In order to test \sawsim, I've developed a test
+suite that compares simulated unfolding force histograms with
+analytical histograms for a number of situations where solving for the
+analytical histogram is possible.
+
+\section{Review of current research}
+
+There are two main approaches to modeling protein domain unfolding
+under tension: Bell's and Kramers'\citep{schlierf06,hummer03,dudko06}.
+Bell introduced his model in the context of cell
+adhesion\citep{bell78}, but it has been widely used to model
+mechanical unfolding in
+proteins\citep{rief97a,carrion-vazquez99b,schlierf06} due to it's
+simplicity and ease of use\citep{hummer03}. Kramers introduced his
+theory in the context of thermally activated barrier crossings, which
+is how we use it here.
+
+
+\subsection{Who's who}
+
+The field of mechanical protein unfolding is developing along three main branches.
+Some groups are predominantly theoretical,
+\begin{itemize}
+ \item Evans, University of British Columbia (Emeritus) \\
+ \url{http://www.physics.ubc.ca/php/directory/research/fac-1p.phtml?entnum=55}
+ \item Thirumalai, University of Maryland \\
+ \url{http://www.marylandbiophysics.umd.edu/}
+ \item Onuchic, University of California, San Diego \\
+ \url{http://guara.ucsd.edu/}
+ \item Hyeon, Chung-Ang University (Onuchic postdoc, Thirumalai postdoc?) \\
+ \url{http://physics.chem.cau.ac.kr/} \\
+ \item Dietz (Rief grad) \\
+ \url{http://www.hd-web.de/}
+ \item Hummer and Szabo, National Institute of Diabetes and Digestive and Kidney Diseases \\
+ \url{http://intramural.niddk.nih.gov/research/faculty.asp?People_ID=1615}
+ \url{http://intramural.niddk.nih.gov/research/faculty.asp?People_ID=1559}
+\end{itemize}
+and the experimentalists are usually either AFM based
+\begin{itemize}
+ \item Rief, Technischen Universität München \\
+ \url{http://cell.e22.physik.tu-muenchen.de/gruppematthias/index.html}
+ \item Fernandez, Columbia University \\
+ \url{http://www.columbia.edu/cu/biology/faculty/fernandez/FernandezLabWebsite/}
+ \item Oberhauser, University of Texas Medical Branch (Fernandez postdoc) \\
+ \url{http://www.utmb.edu/ncb/Faculty/OberhauserAndres.html}
+ \item Marszalek, Duke University (Fernandez postdoc) \\
+ \url{http://smfs.pratt.duke.edu/homepage/lab.htm}
+ \item Guoliang Yang, Drexel University \\
+ \url{http://www.physics.drexel.edu/~gyang/}
+ \item Wojcikiewicz, University of Miami \\
+ \url{http://chroma.med.miami.edu/physiol/faculty-wojcikiewicz_e.htm}
+\end{itemize}
+or laser-tweezers based
+\begin{itemize}
+ \item Bustamante, University of California, Berkley \\
+ \url{http://alice.berkeley.edu/}
+ \item Forde, Simon Fraser University \\
+ \url{http://www.sfu.ca/fordelab/index.html}
+\end{itemize}
+
+\subsection{Evolution of unfolding modeling}
+
+Evans introduced the saddle-point Kramers' approximation in a protein unfolding context 1997 (\citet{evans97} Eqn.~3).
+However, early work on mechanical unfolding focused on the simper Bell model\citep{rief97a}.%TODO
+In the early `00's, the saddle-point/steepest-descent approximation to Kramer's model (\citet{hanggi90} Eqn.~4.56c) was introduced into our field\citep{dudko03,hyeon03}.%TODO
+By the mid `00's, the full-blown double-integral form of Kramer's model (\citet{hanggi90} Eqn.~4.56b) was in use\citep{schlierf06}.%TODO
+
+There have been some tangential attempts towards even fancier models.
+\citet{dudko03} attempted to reduce the restrictions of the single-unfolding-path model.
+\citet{hyeon03} attempted to measure the local roughness using temperature dependent unfolding.
+
+\subsection{History of simulations}
+
+Early molecular dynamics (MD) work on receptor-ligand breakage by Grubmuller 1996 and Izrailev 1997 (according to Evans 1997).
+\citet{evans97} introduce a smart Monte Carlo (SMC) Kramers' simulation.
+
+\subsection{History of experimental AFM unfolding experiments}
+
+\begin{itemize}
+ \item \citet{rief97a}:
+\end{itemize}
+
+\subsection{History of experimental laser tweezer unfolding experiments}
+
+\begin{itemize}
+ \item \citet{izrailev97}:
+\end{itemize}
+
+\section{Single-domain proteins under constant loading}
+
+eq:sawsim:order-dep
+
+Let $x$ be the end to end distance of the protein, $t$ be the time since loading began, $F$ be tension applied to the protein, $P$ be the surviving population of folded proteins.
+Make the definitions
+\begin{align}
+ v &\equiv \deriv{t}{x} && \text{the pulling velocity} \\
+ k &\equiv \deriv{x}{F} && \text{the loading spring constant} \\
+ P_0 &\equiv P(t=0) && \text{the initial number of folded proteins} \\
+ D &\equiv P_0 - P && \text{the number of dead (unfolded) proteins} \\
+ \kappa &\equiv -\frac{1}{P} \deriv{t}{P} && \text{the unfolding rate}
+\end{align}
+\nomenclature{$\equiv$}{Defined as (\ie equivalent to)}
+The proteins are under constant loading because
+\begin{equation}
+ \deriv{t}{F} = \deriv{x}{F}\deriv{t}{x} = kv\;,
+\end{equation}
+a constant, since both $k$ and $v$ are constant (\citet{evans97} in the text on the first page, \citet{dudko06} in the text just before Eqn.~4).
+
+The instantaneous likelyhood of a protein unfolding is given by $\deriv{F}{D}$, and the unfolding histogram is merely this function discretized over a bin of width $W$(This is similar to \citet{dudko06} Eqn.~2, remembering that $\dot{F}=kv$, that their probability density is not a histogram ($W=1$), and that their pdf is normalized to $N=1$).
+\begin{equation}
+ h(F) \equiv \deriv{\text{bin}}{F}
+ = \deriv{F}{D} \cdot \deriv{\text{bin}}{F}
+ = W \deriv{F}{D}
+ = -W \deriv{F}{P}
+ = -W \deriv{t}{P} \deriv{F}{t}
+ = \frac{W}{vk} P\kappa \label{eq:unfold:hist}
+\end{equation}
+Solving for theoretical histograms is merely a question of taking your chosen $\kappa$, solving for $P(f)$, and plugging into Eqn. \ref{eq:unfold:hist}.
+We can also make a bit of progress solving for $P$ in terms of $\kappa$ as follows:
+\begin{align}
+ \kappa &\equiv -\frac{1}{P} \deriv{t}{P} \\
+ -\kappa \dd t \cdot \deriv{t}{F} &= \frac{\dd P}{P} \\
+ \frac{-1}{kv} \int \kappa \dd F &= \ln(P) + c \\
+ P &= C\exp{\p({\frac{-1}{kv}\integral{}{}{F}{\kappa}})} \;, \label{eq:P}
+\end{align}
+where $c \equiv \ln(C)$ is a constant of integration scaling $P$.
+
+\subsection{Constant unfolding rate}
+
+In the extremely weak tension regime, the proteins' unfolding rate is independent of tension, we have
+\begin{align}
+ P &= C\exp{\p({\frac{-1}{kv}\integral{}{}{F}{\kappa}})}
+ = C\exp{\p({\frac{-1}{kv}\kappa F})}
+ = C\exp{\p({\frac{-\kappa F}{kv}})} \\
+ P(0) &\equiv P_0 = C\exp(0) = C \\
+ h(F) &= \frac{W}{vk} P \kappa
+ = \frac{W\kappa P_0}{vk} \exp{\p({\frac{-\kappa F}{kv}})}
+\end{align}
+So, a constant unfolding-rate/hazard-function gives exponential decay.
+Not the most earth shattering result, but it's a comforting first step, and it does show explicitly the dependence in terms of the various unfolding-specific parameters.
+
+\subsection{Bell model}
+
+Stepping up the intensity a bit, we come to Bell's model for unfolding
+(\citet{hummer03} Eqn.~1 and the first paragraph of \citet{dudko06} and \citet{dudko07}).
+\begin{equation}
+ \kappa = \kappa_0 \cdot \exp\p({\frac{F \dd x}{k_B T}})
+ = \kappa_0 \cdot \exp(a F) \;,
+\end{equation}
+where we've defined $a \equiv \dd x/k_B T$ to bundle some constants together.
+The unfolding histogram is then given by
+\begin{align}
+ P &= C\exp\p({\frac{-1}{kv}\integral{}{}{F}{\kappa}})
+ = C\exp\p[{\frac{-1}{kv} \frac{\kappa_0}{a} \exp(a F)}]
+ = C\exp\p[{\frac{-\kappa_0}{akv}\exp(a F)}] \\
+ P(0) &\equiv P_0 = C\exp\p({\frac{-\kappa_0}{akv}}) \\
+ C &= P_0 \exp\p({\frac{\kappa_0}{akv}}) \\
+ P &= P_0 \exp\p\{{\frac{\kappa_0}{akv}[1-\exp(a F)]}\} \\
+ h(F) &= \frac{W}{vk} P \kappa
+ = \frac{W}{vk} P_0 \exp\p\{{\frac{\kappa_0}{akv}[1-\exp(a F)]}\} \kappa_0 \exp(a F)
+ = \frac{W\kappa_0 P_0}{vk} \exp\p\{{a F + \frac{\kappa_0}{akv}[1-\exp(a F)]}\} \label{eq:unfold:bell_pdf}\;.
+\end{align}
+The $F$ dependent behavior reduces to
+\begin{equation}
+ h(F) \propto \exp\p[{a F - b\exp(a F)}] \;,
+\end{equation}
+where $b \equiv \kappa_0/akv \equiv \kappa_0 k_B T / k v \dd x$ is
+another constant rephrasing.
+
+This looks similar to the Gompertz / Gumbel / Fisher-Tippett
+distribution, where
+\begin{align}
+ p(x) &\propto z\exp(-z) \\
+ z &\equiv \exp\p({-\frac{x-\mu}{\beta}}) \;,
+\end{align}
+but we have
+\begin{equation}
+ p(x) \propto z\exp(-bz) \;.
+\end{equation}
+Strangely, the Gumbel distribution is supposed to derive from an
+exponentially increasing hazard function, which is where we started
+for our derivation. I haven't been able to find a good explaination
+of this discrepancy yet, but I have found a source that echos my
+result (\citet{wu04} Eqn.~1). TODO: compare \citet{wu04} with
+my successful derivation in \cref{sec:sawsim:results-scaffold}.
+
+Oh wait, we can do this:
+\begin{equation}
+ p(x) \propto z\exp(-bz) = \frac{1}{b} z'\exp(-z')\propto z'\exp(-z') \;,
+\end{equation}
+with $z'\equiv bz$. I feel silly... From
+\href{Wolfram}{http://mathworld.wolfram.com/GumbelDistribution.html},
+the mean of the Gumbel probability density
+\begin{equation}
+ P(x) = \frac{1}{\beta} \exp\p[{\frac{x-\alpha}{\beta}
+ -\exp\p({\frac{x-\alpha}{\beta}})
+ }]
+\end{equation}
+is given by $\mu=\alpha-\gamma\beta$, and the variance is
+$\sigma^2=\frac{1}{6}\pi^2\beta^2$, where $\gamma=0.57721566\ldots$ is
+the Euler-Mascheroni constant. Selecting $\beta=1/a=k_BT/\dd x$,
+$\alpha=-\beta\ln(\kappa\beta/kv)$, and $F=x$ we have
+\begin{align}
+ P(F)
+ &= \frac{1}{\beta} \exp\p[{\frac{F+\beta\ln(\kappa\beta/kv)}{\beta}
+ -\exp\p({\frac{F+\beta\ln(\kappa\beta/kv)}
+ {\beta}})
+ }] \\
+ &= \frac{1}{\beta} \exp(F/\beta)\exp[\ln(\kappa\beta/kv)]
+ \exp\p\{{-\exp(F/\beta)\exp[\ln(\kappa\beta/kv)]}\} \\
+ &= \frac{1}{\beta} \frac{\kappa\beta}{kv} \exp(F/\beta)
+ \exp\p[{-\kappa\beta/kv\exp(F/\beta)}] \\
+ &= \frac{\kappa}{kv} \exp(F/\beta)\exp[-\kappa\beta/kv\exp(F/\beta)] \\
+ &= \frac{\kappa}{kv} \exp(F/\beta - \kappa\beta/kv\exp(F/\beta)] \\
+ &= \frac{\kappa}{kv} \exp(aF - \kappa/akv\exp(aF)] \\
+ &= \frac{\kappa}{kv} \exp(aF - b\exp(aF)]
+ \propto h(F) \;.
+\end{align}
+So our unfolding force histogram for a single Bell domain under
+constant loading does indeed follow the Gumbel distribution.
+
+% Consolidate with src/sawsim/discussion.tex
+
+\subsection{Saddle-point Kramers' model}
+
+For the saddle-point approximation for Kramers' model for unfolding
+(\citet{evans97} Eqn.~3, \citet{hanggi90} Eqn. 4.56c, \citet{vanKampen07} Eqn. XIII.2.2).
+\begin{equation}
+ \kappa = \frac{D}{l_b l_{ts}} \cdot \exp\p({\frac{-E_b(F)}{k_B T}}) \;,
+\end{equation}
+where $E_b(F)$ is the barrier height under an external force $F$,
+$D$ is the diffusion constant of the protein conformation along the reaction coordinate,
+$l_b$ is the characteristic length of the bound state $l_b \equiv 1/\rho_b$,
+$\rho_b$ is the density of states in the bound state, and
+$l_{ts}$ is the characteristic length of the transition state
+\begin{equation}
+ l_{ts} = TODO
+\end{equation}
+
+\citet{evans97} solved this unfolding rate for both inverse power law potentials and cusp potentials.
+
+\subsubsection{Inverse power law potentials}
+
+\begin{equation}
+ E(x) = \frac{-A}{x^n}
+\end{equation}
+(e.g. $n=6$ for a van der Waals interaction, see \citet{evans97} in
+the text on page 1544, in the first paragraph of the section
+\emph{Dissociation under force from an inverse power law attraction}).
+Evans then goes into diffusion constants that depend on the
+protein's end to end distance, and I haven't worked out the math
+yet. TODO: clean up.
+
+
+\subsubsection{Cusp potentials}
+
+\begin{equation}
+ E(x) = \frac{1}{2}\kappa_a \p({\frac{x}{x_a}})^2
+\end{equation}
+(see \citet{evans97} in the text on page 1545, in the first paragraph
+of the section \emph{Dissociation under force from a deep harmonic well}).
+
+\section{Double-integral Kramers' theory}
+
+The double-integral form of overdamped Kramers' theory may be too
+complex for analytical predictions of unfolding-force histograms.
+Rather than testing the entire \sawsim\ simulation (\cref{sec:sawsim}),
+we will focus on demonstrating that the Kramers' $k(F)$ evaluations
+are working properly. If the Bell modeled histograms check out, that
+gives reasonable support for the $k(F) \rightarrow \text{histogram}$
+portion of the simulation.
+
+Looking for analytic solutions to Kramers' $k(F)$, we find that there
+are not many available in a closed form. However, we do have analytic
+solutions for unforced $k$ for cusp-like and quartic potentials.
+
+\subsection{Cusp-like potentials}
+
+
+\subsection{Quartic potentials}
-\section{Introduction}
-\label{sec:sawsim:introduction}
-
% AFM unfolding analysis, what we'll do.
Much theoretical and computational work has been done in order to
extract information about the structural, kinetic, and energetic
Hookean force of the bent cantilever. The unfolding probability of
the protein molecules in the polymer is then calculated for that
tension, and whether an unfolding event occurs is determined according
-to a Monte Carlo method. The simulation was implemented in
-C\footnote{Source code available at
- \url{http://www.physics.drexel.edu/~wking/sawsim/}}.
+to a Monte Carlo method. The simulation was implemented in C, and
+there are a number of Python modules to facilitate running several
+simulations in parallel\footnote{Source code available at
+ \url{http://blog.tremily.us/posts/sawsim/}}.
-\subsection{Generating force curves}
-\label{sec:sawsim:methods-tension}
+In the following sections, we'll discuss models used to determine the
+tension of a chain composed of several types of ``domains'' (e.g.~one
+cantilever, three folded I27 domains, and seven unfolded I27 domains)
+(\cref{sec:sawsim:tension}). We'll also work through a number of
+models for calculating the probability that a domain will transition
+from one state (e.g.~folded I27) to another (e.g.~unfolded I27)
+(\cref{sec:sawsim:rate}).
+
+\subsection{Modeling polymer tension}
+\label{sec:sawsim:tension}
% introduce domains and groups.
The fundamental abstraction of the simulation is the ``domain'', which
represents a discrete chunk of the flexible chain between the
-substrate and the cantilever holder. Each of these domains is
-assigned a particular state; for example, the domain representing the
-cantilever is assigned to the ``cantilever'' state, and the domains
-representing protein molecules are assigned to either the ``folded''
-or the ``unfolded'' state. When balancing the tension along the
-chain, we assume that the spatial order of domains along the chain is
-irrelevant\citep{li00}, and therefore, the domains can be rearranged
-and grouped by state. To determine the tension in the chain and the
-amount of cantilever bending when $n$ states are populated, a system
-of $n+1$ equations with $n+1$ unknowns must be solved
+substrate and the cantilever holder (\cref{fig:sawsim:domain-chain}).
+Each of these domains is assigned a particular state; for example, the
+domain representing the cantilever is assigned to the ``cantilever''
+state, and the domains representing protein molecules are assigned to
+either the ``folded'' or the ``unfolded'' state. When balancing the
+tension along the chain, we assume that the spatial order of domains
+along the chain is irrelevant\citep{li00}, so the domains can be
+rearranged and grouped by state (\cref{fig:sawsim:domain-states}). To
+determine the tension in the chain and the amount of cantilever
+bending when $N$ states are populated, a system of $N+1$ equations
+with $N+1$ unknowns must be solved
\begin{align}
F_i(x_i) &= F_t \label{eq:sawsim:tension-balance} \\
\sum_i x_i &= x_t \;, \label{eq:sawsim:x-total}
\end{align}
where $F$ are tensions, $x$ are extensions, and the subscripts $i$ and
$t$ represent a particular state group and the total chain
-respectively (\cref{fig:unfolding-schematic}). From this $F(x_t)$ may be
-computed using any multi-dimensional root-finding algorithm.
+respectively (\cref{fig:unfolding-schematic}). $F(x_t)$ may be
+computed from this system of equations using any multi-dimensional
+root-finding algorithm.
+
+\begin{figure}
+ \begin{center}
+ \subfloat[][]{\label{fig:sawsim:domain-chain}
+ \begin{tikzpicture}[->,node distance=2cm,font=\footnotesize]
+ \tikzstyle{every state}=[fill,draw=red!50,very thick,fill=red!20]
+ \node[state] (A) {domain 1};
+ \node[state] (B) [below of=A] {domain 2};
+ \node[state] (C) [below of=B] {.~.~.};
+ \node[state] (D) [below of=C] {domain $N$};
+ \node (S) [below of=D] {Surface};
+ \node (E) [above of=A] {};
+
+ \path[-] (A) edge (B)
+ (B) edge node [right] {Tension} (C)
+ (C) edge (D)
+ (D) edge (S);
+ \path[->,green] (A) edge node [right,black] {Extension} (E);
+ \end{tikzpicture}}
+ \hspace{.25in}%
+ \subfloat[][]{\label{fig:sawsim:domain-states}
+ \begin{tikzpicture}[->,node distance=2.5cm,shorten <=1pt,shorten >=1pt,font=\footnotesize]
+ \tikzstyle{every state}=[fill,draw=blue!50,very thick,fill=blue!20]
+ \node[state] (A) {cantilever};
+ \node[state] (C) [below of=A] {transition};
+ \node[state] (B) [left of=C] {folded};
+ \node[state] (D) [right of=C] {unfolded};
+
+ \path (B) edge [bend left] node [above] {$k_1$} (C)
+ (C) edge [bend left] node [below] {$k_1'$} (B)
+ edge [bend left] node [above] {$k_2$} (D)
+ (D) edge [bend left] node [below] {$k_2'$} (C);
+ \end{tikzpicture}}
+ \caption{\subref{fig:sawsim:domain-chain} Extending a chain of
+ domains. One end of the chain is fixed, while the other is
+ extended at a constant speed. The domains are coupled with
+ rigid linkers, so the domains themselves must stretch to
+ accomodate the extension. Compare with
+ \cref{fig:unfolding-schematic}.
+ \subref{fig:sawsim:domain-states} Each domain exists in a
+ discrete state. At each timestep, it may transition into
+ another state following a user-defined state matrix such as this
+ one, showing a metastable transition state and an explicit
+ ``cantilever'' domain.\label{fig:sawsim:domains}}
+ \end{center}
+\end{figure}
+
+\subsubsection{Hooke's law}
+\label{sec:sawsim:tension:hooke}
% introduce particular models, and mention parameter aggregation
Inside this framework, we chose a particular extension model
$F_i(x_i)$ for each domain state. Cantilever elasticity is described
by Hooke's law, which gives
+\index{Hooke's law}
\begin{equation}
F = \kappa_c x_c \;, \label{eq:sawsim:hooke}
\end{equation}
where $\kappa_c$ is the bending spring constant and $x_c$ is the
deflection of the cantilever (\cref{fig:unfolding-schematic}).
-Unfolded domains are modeled as WLCs (\cref{sec:tension:wlc}).
-The chain of $N_f$ folded domains is modeled as a string, free to
-assume any extension up to some fixed contour length $L_f=N_fL_{f1}$
+\subsubsection{Wormlike chains}
+\label{sec:sawsim:tension:wlc}
+
+\index{Wormlike chains}
+Unfolded domains are modeled as wormlike chains
+(WLCs)\citep{rief97a,carrion-vazquez00}, which treat the unfolded
+polymer as an elastic rod of persistence length $p$ and contour length
+$L$ (\cref{fig:wlc}). The relationship between tension $F$ and
+extension (end-to-end distance) $x$ is given by Bustamante's
+interpolation formula\citep{marko95,bustamante94}.
+\nomenclature{WLC}{Wormlike chain, an entropic spring model}
+\begin{equation}
+ F_\text{WLC}(x,p,L) = \frac{k_B T}{p}
+ \p[{ \frac{1}{4}\p({ \frac{1}{(1-x/L)^2} - 1 })
+ + \frac{x}{L} }] \;,
+ \label{eq:sawsim:wlc}
+\end{equation}
+where $p$ is the persistence length. This interpolation forumla is
+accurate to within 7\% of the exact $F_\text{WLC}$ for
+$F_\text{WLC}\approx k_B T/p_u$\citep{marko95}. Because most unfolded
+proteins studied have persistence lengths on the order of the size of
+an amino acid
+($p_u\approx3.8\U{\AA}$\citep{rief97a,carrion-vazquez99b,carrion-vazquez00}),
+this characteristic force works out to be around $11\U{pN}$. Most
+proteins studied using force spectroscopy have unfolding forces in the
+hundreds of piconewtons, by which point the interpolation formula is
+in it's more accurate high-extension regime.
+\nomenclature{\AA}{{\AA}ngstr{\"o}m, a unit of length.
+ $1\U{\AA}=1\E{-10}\U{m}$}
+
+For chain with $N_u$ unfolded domains sharing a persistence length
+$p_u$ and per-domain contour lengths $L_{u1}$, the tension of the WLC
+is determine by summing the contour lengths
+\begin{equation}
+ F(x, p_u, L_u, N_u) = F_\text{WLC}(x, p_u, N_u L_{u1}) \;.
+ \label{eq:sawsim:multi-wlc}
+\end{equation}
+
+\begin{figure}
+ \begin{center}
+ \subfloat[][]{\asyinclude{figures/schematic/wlc-model}%
+ \label{fig:wlc-model}}
+ \hspace{.25in}%
+ \subfloat[][]{\asyinclude{figures/schematic/wlc-extension}%
+ \label{fig:wlc-extension}}
+ \caption{\subref{fig:wlc-model} The wormlike chain models a
+ polymer as an elastic rod with persistence length $p$ and
+ contour length $L$. \subref{fig:wlc-extension} Force
+ vs.~extension for a WLC using Bustamante's interpolation
+ formula.\label{fig:wlc}}
+ \end{center}
+\end{figure}
+
+\subsubsection{Folded domains}
+\label{sec:sawsim:tension:folded}
+
+A short chain of folded proteins, however, are not easily described by
+polymer models. Several studies have used WLC and FJC models to fit
+the elastic properties of the modular protein
+titin\citep{granzier97,linke98a},
+% TODO: check it really is folded domains \& bulk titin
+but native titin contains hundreds of folded and unfolded domains.
+For the short protein polymers common in mechanical unfolding
+experiments (\cref{sec:polymer-synthesis}), the cantilever dominates
+the elasticity of the polymer-cantilever system before any protein
+molecules unfold. After the first unfolding event occurs, the
+unfolded portion of the chain is already longer and softer than the
+sum of all the remaining folded domains, and dominates the elasticity
+of the whole chain. Therefore, the details of the tension model
+chosen for the folded domains has negligible effect on the unfolding
+forces (\cref{eq:sawsim:x-total}), which was also suggested by
+\citet{staple08}. Force curves simulated using different models to
+describe the folded domains yielded almost identical unfolding force
+distributions (data not shown, TODO: show data).
+
+As an alternative to modeling the folded domains explicitly or
+ignoring them completely, another approach is to subtract the
+end-to-end length of the folded protein from the contour length of the
+unfolded protein to create an effective contour length for the
+unfolding\cite{carrion-vazquez99b}. This effectivly models the folded
+domains as WLCs with the same persistence length as the unfolded
+domains.
+
+%The chain of $N_f$ folded domains is modeled as a string, free to
+%assume any extension up to some fixed contour length $L_f=N_fL_{f1}$
+%\begin{equation}
+% F = \begin{cases}
+% 0 & \text{if $x_f<L_f$} \;, \\
+% \infty & \text{if $x_f>L_f$} \;,
+% \end{cases} \label{eq:sawsim:piston}
+%\end{equation}
+%where $L_{f1}$ is the separation of the two linking points of a folded
+%domain, and $x_f$ is the end-to-end length of the chain of folded
+%domains. In this model, any non-zero tension will fully extend these
+%folded domains. Because the range of possible extensions for folded
+%domains is so short, the contribution of the folded domains to the
+%elastic behavior of the polymer-cantilever system is relatively
+%insignificant.
+
+\subsubsection{Other models}
+\label{sec:sawsim:tension:other}
+
+\index{FJC}
+\index{Kuhn length}
+The unfolded polypeptide chain has been shown to follow the WLC model
+quite well\citep{rief97a} (\cref{sec:sawsim:tension:wlc}), though
+other polymer models have been tried. One alternative is the
+freely-jointed chain
+(FJC)\citep{kellermayer97,linke98a,janshoff00,verdier70}, which models
+the polymer as a series of $N$ rigid links, each of length $l$ (the
+Kuhn length), which are free to rotate about their joints
+(\cref{fig:fjc}).
+\index{Langevin function}
+\nomenclature{FJC}{Freely-jointed chain, an entropic spring model}
+\nomenclature{$\Langevin$}{The Langevin function,
+ $\Langevin(\alpha)\equiv\coth{\alpha}-\frac{1}{\alpha}$}
\begin{equation}
- F = \begin{cases}
- 0 & \text{if $x_f<L_f$} \;, \\
- \infty & \text{if $x_f>L_f$} \;,
- \end{cases} \label{eq:sawsim:piston}
+ F_\text{FJC}(x,l,L) = \frac{k_B T}{l} \Langevin^{-1}\p({\frac{x}{L}}) \;,
+ \label{eq:sawsim:fjc}
\end{equation}
-where $L_{f1}$ is the separation of the two linking points of a folded
-domain, and $x_f$ is the end-to-end length of the chain of folded
-domains. In this model, any non-zero tension will fully extend these
-folded domains. As discussed in \cref{sec:tension:folded}, the
-contribution of the folded domains to the elastic behavior of the
-polymer-cantilever system is relatively insignificant.
+where $L=Nl$ is the total length of the chain, and
+$\Langevin(\alpha)\equiv\coth{\alpha}-\frac{1}{\alpha}$ is the
+Langevin function\citep{hatfield99}.
+
+\begin{figure}
+ \begin{center}
+ \subfloat[][]{\asyinclude{figures/schematic/fjc-model}%
+ \label{fig:fjc-model}}
+ \hspace{.25in}%
+ \subfloat[][]{\asyinclude{figures/schematic/fjc-extension}%
+ \label{fig:fjc-extension}}
+ \caption{\subref{fig:fjc-model} The freely-jointed chain models
+ the polymer as a series of $N$ rigid links, each of length $l$,
+ which are free to rotate about their joints. Each polymer state
+ is a random walk, and the density of states for a given
+ end-to-end distance is determined by the number of random walks
+ that have such an end-to-end distance.
+ \subref{fig:fjc-extension} Force vs.~extension for a
+ hundred-segment FJC. The WLC extension curve (with $p=l$) is
+ shown as a dashed line for comparison.\label{fig:fjc}}
+ \end{center}
+\end{figure}
+
+More exotic models such as elastic WLCs\citep{janshoff00,puchner08},
+elastic FJCs\citep{fisher99a,janshoff00}, and freely rotating
+chains\cite{puchner08} (FRCs) have also been used to model DNA and
+polysaccharides, but are rarely used to model the relatively short and
+inextensible synthetic proteins used in force spectroscopy.
+\nomenclature{FRC}{Freely-rotating chain (like the FJC, except that
+ the bond angles are fixed. The torsional angles are not
+ restricted)}
+
+
+\subsubsection{Assumptions}
+\label{sec:sawsim:tension:assumptions}
% address assumptions & caveats
In the simulation, the protein polymer is assumed to be stretched in
length of a protein molecule is much larger than that of the folded
form. Therefore, after one molecule unfolds, the polymer becomes much
longer and the angle between the polymer and the surface approaches
-$90$ degrees\citep{carrion-vazquez00}. The joints between domain
-groups are assumed to lie along a line between the surface tether
-point and the position of the tip (\cref{eq:sawsim:x-total}). The
+$90$ degrees\citep{carrion-vazquez00}.
+
+The joints between domain groups are also assumed to lie along a line
+between the surface tether point and the position of the tip
+(\cref{eq:sawsim:x-total} is scalar, not vector, addition). The
effects of this assumption are also minimized due to greater length of
-the unfolded domain. Finally, the interactions between different
-parts of the polymer and between the chain and the surface (except at
-the tethering points) are not considered. This is reasonable since
-these interactions should not make substantial contributions to the
-force curve at the force levels of interest, where the polymer is in a
-relatively extended conformation.
+the unfolded domain compared with the other domains (folded proteins
+and cantilever deflection). For example, a $0.050\U{N/m}$ cantilever
+under $200\U{pN}$ of tension will bend $x_c=F/\kappa_c=4\U{nm}$. The
+entire end-to-end length of folded domains such as I27 are also around
+$5\U{nm}$ (\cref{fig:I27}). A single unfolded I27, with its 89 amino
+acids\citep{improta96}, should have an unfolded contour length of
+$89\U{aa}\cdot0.38\U{nm}=33.8\U{nm}$, equivalent to a cantilever and
+five folded domains.
+
+\subsubsection{Velocity-clamp example}
+\label{sec:sawsim:velocity-clamp}
% introduce constant velocity and walk through explicit example pull
-Consider an experiment of pulling a polymer with $N$ identical protein
-molecules at a constant speed. At the start of an experiment, the
+Consider an experiment pulling a polymer with $N$ identical protein
+domains at a constant speed. At the start of an experiment, the
chain is unstretched ($x_t=0$), which means all the domains are
unstretched, the cantilever is undeflected, and the tip is in contact
with the surface. There is one domain in the cantilever state, $N$ in
The simulation assumes that the pulling takes discrete steps in space
and treats $x_t$ as constant over the duration of one time step
$\Delta t$. Because of the adaptive time steps discussed in
-\cref{sec:sawsim:methods-timesteps}, the space steps $\Delta x_t = v\Delta t$
+\cref{sec:sawsim:timesteps}, the space steps $\Delta x_t = v\Delta t$
may have different sizes, but each step will be ``small''. At each
-step, the total extension is calculated using \cref{eq:sawsim:const-v}, and
-the tension $F(x_t=vt)$ is determined by numerically solving
-\cref{eq:sawsim:tension-balance,eq:sawsim:x-total} using the models
-\cref{eq:sawsim:hooke,eq:sawsim:wlc,eq:sawsim:piston} for known values of the parameters in
-the various states $(N_u, N_f, v, \kappa_c, L_{f1}, L_{u1}, p_u)$.
-When one of the molecules in the polymer unfolds
-(\cref{sec:sawsim:methods-unfolding}), there will be one domain in the
-unfolded state and $N-1$ in the folded state. In the next step, a
-newly balanced tension between the cantilever and the polymer is
-determined by solving for $F(x_t)$ as discussed above, but with the
-total extension $x_t$ incremented by $v\Delta t$ and the new unfolded
-contour length $L_{u1}$ and folded contour length $(N-1)L_{f1}$. The
-sudden lengthening of the polymer chain results in a corresponding
-abrupt drop in the force, leading to the formation of one sawtooth in
-the force curve. As the pulling continues and more domains unfold,
-force curves with a series of sawteeth are generated
+step, the total extension is calculated using
+\cref{eq:sawsim:const-v}, and the tension $F(x_t=vt)$ is determined by
+numerically solving \cref{eq:sawsim:tension-balance,eq:sawsim:x-total}
+using the models \cref{eq:sawsim:hooke,eq:sawsim:wlc}
+%,eq:sawsim:piston}
+for known values of the parameters in the various states $(N_u, N_f,
+v, \kappa_c,
+% L_{f1},
+L_{u1}, p_u)$. When one of the molecules in the
+polymer unfolds (\cref{sec:sawsim:rate}), there will be
+one domain in the unfolded state and $N-1$ in the folded state. In
+the next step, a newly balanced tension between the cantilever and the
+polymer is determined by solving for $F(x_t)$ as discussed above, but
+with the total extension $x_t$ incremented by $v\Delta t$ and the new
+unfolded contour length $L_{u1}$ and folded contour length
+$(N-1)L_{f1}$. The sudden lengthening of the polymer chain results in
+a corresponding abrupt drop in the force, leading to the formation of
+one sawtooth in the force curve. As the pulling continues and more
+domains unfold, force curves with a series of sawteeth are generated
(\cref{fig:sawsim:sim-sawtooth}).
+\subsubsection{Equlibration timescales}
+\label{sec:sawsim:timescales}
+
The tension calculation assumes an equilibrated chain, so
consideration must be given to the chain's relaxation time, which
should be short compared to the loading timescale. The relaxation
time for a WLC\index{WLC!relaxation time} is given by
\begin{equation}
\tau \approx \eta \frac{k_BT p}{F^2}
-% < \eta \frac{k_BT p}{(k_BTx/pL)^2} =
+% < \eta \frac{k_BT p}{(k_BTx/pL)^2}
% Note: < because F > k_BTx/pL
% < \frac{\eta p^3 L^2}{k_BT x^2}
% < \frac{\eta p^2 L}{k_BT} % for x/L > \sqrt{p/L}
the persistence length\citep{evans99}. For forces greater than
$1\U{pN}$, with $\eta_\text{water}/k_BT=2.45\E{-10}\U{s/nm$^3$}$,
$\tau<2\U{ns}$ for the protein polymer used in the simulation.
-% python -c 'print 2.45e-10*(1e9)**3 * (1.38e-23*300)**2 * 0.4e-9 / (1e-12)**2'
+% eta/(k_BT) * (k_B *T )**2 * p / F**2
+% python -c 'print(2.45e-10*(1e9)**3 * (1.38e-23*300)**2 * 0.4e-9 / (1e-12)**2)'
+% s/m**3 * (J/K *K )**2 * m / N**2 = s
% 1.68...e-09
Therefore, the polymer chain is equilibrated almost instantaneously
within a time step, which is on the order of tens of microseconds.
the force measurement.
\subsection{Unfolding protein molecules by force}
-\label{sec:sawsim:methods-unfolding}
+\label{sec:sawsim:rate}
+
+In the previous section, we discussed methods for calculating the
+tension of a chain composed of several domains in series
+(\cref{fig:sawsim:domain-chain}). Those methods allow us to calculate
+the tension of the chain for any given extension. We use that tension
+to calculate transition rates between states
+(\cref{fig:sawsim:domain-states}). In this section, we'll introduce
+the Bell model for unfolding (\cref{sec:sawsim:rate:bell}) and mention
+a few more exotic models. We'll wrap up by pointing out some of the
+approximations and assumptions we make when we use these simple models
+(\cref{sec:sawsim:rate:assumptions}).
+
+\subsubsection{Bell model}
+\label{sec:sawsim:rate:bell}
+\index{Bell model}
% introduce Bell, probability calculations, and MC comparison
According to the theory developed by \citet{bell78} and extended by
\citet{evans99}, an external stretching force $F$ increases the
unfolding rate constant of a protein molecule
+\index{Bell model}
\begin{equation}
k_u = k_{u0} \exp\p({\frac{F\Delta x_u}{k_B T}}) \;, \label{eq:sawsim:bell}
\end{equation}
where $k_{u0}$ is the unfolding rate in the absence of an external
force, and $\Delta x_u$ is the distance between the native state and
-the transition state along the pulling direction. The probability for
-a protein molecule to unfold under an applied force is
+the transition state along the pulling direction.
+
+\begin{figure}
+ \asyinclude{figures/schematic/landscape-bell}
+ \caption{Energy landscape schematic for Bell model unfolding
+ (\cref{eq:sawsim:bell}), which models folded domains as two-state
+ systems parameterized by an unforced unfolding rate $k_{u0}$ and a
+ distance $\Delta x$ between the folded and transition
+ states.\label{fig:bell-landscape}}
+\end{figure}
+
+\subsubsection{Monte Carlo transitions}
+\label{sec:sawsim:monte-carlo}
+
+We can use the Bell model (or other models, see
+\cref{sec:sawsim:rate:other}) to calculate the unfolding rate $k_u$ at
+a given force for a single domain. The probability for that single
+protein domain to unfold under applied force is
\begin{equation}
P_1 = k_u \Delta t \;, \label{eq:sawsim:prob-one}
\end{equation}
\end{equation}
where the approximation is valid when $N_fP_1 \ll 1$.
+\nomenclature{$k$}{Rate constant for general state transitions
+ (inverse seconds)}
+\nomenclature{$k_u$}{Unfolding rate constant}
+\nomenclature{$k_{u0}$}{Unforced unfolding rate constant}
+\nomenclature{$\Delta x_u$}{Distance between a domain's native state
+ and the transition state along the pulling direction }
+
\begin{figure}
\asyinclude{figures/schematic/monte-carlo}
\caption{Once the unfolding probability has been caculated, we need
where the polymer is assumed to detach from one of the tethering
surfaces. The cantilever deflection becomes zero after this point.
-% address unfolding models
-Although the Bell model (\cref{eq:sawsim:bell}) is the most widely used
-unfolding model due to its simplicity and its applicability to various
-biopolymers\citep{rief98}, other theoretical models have been proposed
-to interpret mechanical unfolding data. For example,
+\subsubsection{Other models}
+\label{sec:sawsim:rate:other}
+
+Although the Bell model (\cref{eq:sawsim:bell}) is the most widely
+used unfolding model due to its simplicity and its applicability to
+various biopolymers\citep{rief98}, other theoretical models have been
+proposed to interpret mechanical unfolding data. For example,
\citet{schlierf06} used the mechanical unfolding data of the protein
-ddFLN4 to demonstrate that Kramers' diffusion model fit the measured
-unfolding force data better than the Bell model for proteins with
-broad free energy barriers. For proteins with relatively narrow
-folded and transition states, the Bell model provides a good
-approximation.
+ddFLN4 to demonstrate that Kramers' diffusion model (in the
+spatial-diffusion-limited case, a.k.a. the Smoluchowski
+limit)\citep{kramers40,hanggi90,evans97,shillcock98,vanKampen07} fit
+the measured unfolding force data better than the Bell model for
+proteins with broad free energy barriers.
+\index{Kramers' model}
+\index{Diffusion coefficient}
+\index{Free energy}
+\index{Unfolding coordinate}
+\begin{equation}
+ \frac{1}{k_u}
+ = \frac{1}{D}
+ \integral{-\infty}{\infty}{x}{%
+ e^{\frac{U_F(x)}{k_B T}}
+ \integral{-\infty}{x}{x'}{%
+ e^{\frac{-U_F(x')}{k_B T}}}} \;,
+ \label{eq:kramers}
+\end{equation}
+where $D$ is the diffusion coefficient and $U_F(x)$ is the free energy
+along the unfolding cordinate $x$ (\cref{fig:landscape:kramers}).
+\nomenclature{$D$}{Diffusion coefficient (square meters per second)}
+\nomenclature{$U_F(x)$}{Protein free energy along the unfolding
+ coordinate $x$ (joules)}
+
+\begin{figure}
+ \begin{center}
+ \subfloat[][]{\asyinclude{figures/schematic/landscape}%
+ \label{fig:landscape}}
+ % \hspace{.25in}%
+ \subfloat[][]{\asyinclude{figures/schematic/kramers-integrand}%
+ \label{fig:kramers:integrand}}
+ \caption{\subref{fig:landscape} Energy landscape schematic for
+ Kramers integration (compare with \cref{fig:bell-landscape}).
+ \subref{fig:kramers:integrand} A map of the magnitude of
+ Kramers' integrand, with black lines tracing the integration
+ region. The bulk of the contribution to the integral comes from
+ the bump in the upper left, with $x$ near the boundary and $x'$
+ near the folded state. This is why you can calculate a close
+ approximation to this integral by restricting the integration to
+ $x_\text{min}$ and $x_\text{max}$, located a few $k_B T$ beyond
+ the folded and transition states respectively. The restricted
+ integral is much easier to calculate numerically than one bound
+ by $\pm\infty$.
+ (\cref{eq:kramers}).\label{fig:landscape:kramers}}
+ \end{center}
+\end{figure}
+
+When you are simulating the double integral form of Kramers' model,
+you obviously need to parameterize $U_F(x)$ somehow. There has not
+been much research done in this direction, but \citet{schlierf06} used
+cubic splines with 15 variable knots. \citet{shillcock98} used a
+cubic free energy with variable coefficients. The amount of
+information you can extract from fitting such a model to your data is
+limitless, but you run the risk of over-specifying if you add too many
+parameters when you're fitting noisy data.
+
+There are alternative formulations of Kramers' model besides the full
+double integral approach. You can use a Gaussian steepest-descent
+approximation (a.k.a. stationary phase method or saddle-point method)
+to reduce the integral to a formula that only depends on the free
+energy landscape via the curvature $\npderiv{2}{x}{U_F}$ evaluated at
+the folded state and transition state\citep{hanggi90}. This approach
+makes sense for sufficiently sharp folded and transition states, where
+these two measurements will capture the shape of the large-integrand
+region (\cref{fig:kramers:integrand}). The steepest-descent
+formulation has less to say about the underlying energy landscape, but
+it may be more robust in the face of noisy data.
+
+Other tension models in use include a stiffness-corrected
+Bell model\citep{walton08}, and TODO.
+
+How to choose which unfolding model to use? For proteins with
+relatively narrow folded and transition states, the Bell model
+provides a good approximation, and it is the model used by the vast
+majority of earlier work in the field.
+
+\subsubsection{Assumptions}
+\label{sec:sawsim:rate:assumptions}
+
+The interactions between different parts of the polymer and between
+the chain and the surface (except at the tethering points) are often
+ignored. This is usually reasonable since these interactions should
+not make substantial contributions to the force curve at the force
+levels of interest, where the polymer is in a relatively extended
+conformation. However, \citet{li00} showed that while the unfolding
+properties of I27 are not effected by I28 flankers, I28 \emph{is}
+stabilized by neighboring I27. The unforced Bell model unfolding rate
+for I28 in (I28)\textsubscript{8} was $2.8\E{-5}\U{s$^-1$}$, while in
+(I27-I28)\textsubscript{4} it dropped to
+$2.5\E{-6}\U{s$^-1$}$\citep{li00}.
\subsection{Choosing the simulation time steps}
-\label{sec:sawsim:methods-timesteps}
+\label{sec:sawsim:timesteps}
The demands on the time step vary throughout a simulated pull due to
the non-linear elasticity of the polymer. Within a specified time
Within each time step, the total chain extension $x_t$ is treated as a
constant and a force balance is reached very quickly among the various
-domains (see \cref{sec:sawsim:methods-tension} for equilibration timescales).
-This force is used to determine the unfolding probability
-(\cref{eq:sawsim:prob-one,eq:sawsim:prob-n}), which determines the domain state
-populations in the next time step. Therefore, the chain tension must
-not change appreciably over the course of the time step ($\Delta F <
-1\U{pN}$), and the unfolding probability is only calculated once for
-the entire step. The time step must also be short enough that the
-probability of unfolding in a single time step is low ($P<10^{-3}$).
-Besides ensuring that the approximations made in
-\cref{eq:sawsim:prob-one,eq:sawsim:prob-n} are valid, this restriction makes
-time steps which should have multiple unfoldings in a single time step
-highly unlikely. Experimentally measured unfolding are temporally
-separated, because the unfolding transition is characterized by
-multiple, Markovian attempts over a large energy barrier, where the
-probability of crossing the barrier in a single attempt is very low.
-A successful attempt quickly extends the chain contour length,
-reducing the tension, dramatically reducing the likelihood of a second
-escape in that time step. The time step used is recalculated for each
-step so that both of these criteria are satisfied.
+domains (\cref{sec:sawsim:timescales}). This force is used to
+determine the unfolding probability
+(\cref{eq:sawsim:prob-one,eq:sawsim:prob-n}), which determines the
+domain state populations in the next time step. Therefore, the chain
+tension must not change appreciably over the course of the time step
+($\Delta F < 1\U{pN}$), and the unfolding probability is only
+calculated once for the entire step. The time step must also be short
+enough that the probability of unfolding in a single time step is low
+($P<10^{-3}$). Besides ensuring that the approximations made in
+\cref{eq:sawsim:prob-one,eq:sawsim:prob-n} are valid, this restriction
+makes time steps which should have multiple unfoldings in a single
+time step highly unlikely. Experimentally measured unfolding are
+temporally separated, because the unfolding transition is
+characterized by multiple, Markovian attempts over a large energy
+barrier, where the probability of crossing the barrier in a single
+attempt is very low. A successful attempt quickly extends the chain
+contour length, reducing the tension, dramatically reducing the
+likelihood of a second escape in that time step. The time step used
+is recalculated for each step so that both of these criteria are
+satisfied.
+++ /dev/null
-\section{Folded domain tension}
-\label{sec:tension:folded}
-
-The unfolded polypeptide chain has been shown to follow the
-WLC\index{WLC} model quite well (\cref{sec:tension:wlc}), though other
-polymer models, such as the Freely-Jointed Chain
-(FJC)\citep{verdier70}\index{FJC}\nomenclature{FJC}{Freely-Jointed Chain}
-(\cref{sec:tension:fjc}), can be used to fit the force-extension
-relationship\citep{janshoff00}. A short chain of folded proteins,
-however, cannot be described well by polymer models. Several studies
-have used WLC and FJC models to fit the elastic properties of the
-modular protein titin\citep{granzier97,linke98a},
-% TODO: check it really is folded domains \& bulk titin
-but native titin contains hundreds of folded and unfolded domains.
-For the short protein polymers common in mechanical unfolding
-experiments, the cantilever dominates the elasticity of the
-polymer-cantilever system before any protein molecules unfold. After
-the first unfolding event occurs, the unfolded portion of the chain is
-already longer and softer than the sum of all the remaining folded
-domains, and dominates the elasticity of the whole chain. Therefore,
-the details of the tension model chosen for the folded domains has
-negligible effect on the unfolding forces, which was also suggested by
-\citet{staple08}. Force curves simulated using different models to
-describe the folded domains yielded almost identical unfolding force
-distributions (data not shown, TODO: show data).
+++ /dev/null
-\chapter{Chain Tension}
-\label{sec:tension}
-
-\input{tension/polymer}
-\input{tension/folded}
+++ /dev/null
-\section{Polymer Models}
-
-
-\subsection{Wormlike chains}
-\label{sec:tension:wlc}
-
-The unfolded forms of many domains can be modeled as Worm-Like Chains
-(WLCs)\citep{marko95,bustamante94}
-\index{WLC}\nomenclature{WLC}{Wormlike Chain}, which treats the
-unfolded polymer as an elastic rod of persistence length $p$ and
-contour length $L$. The relationship between tension $F$ and
-extension (end-to-end distance) $x$ is given to within XX\% by
-Bustamante's interpolation formula\citep{marko95,bustamante94}.
-\begin{equation}
- F_\text{WLC}(x,p,L) = \frac{k_B T}{p_u}
- \p[{ \frac{1}{4}\p({ \frac{1}{(1-x/L)^2} - 1 })
- + \frac{x}{L} }] \;,
- \label{eq:sawsim:wlc}
-\end{equation}
-where $p$ is the persistence length.
-
-For chain with $N_u$ unfolded domains sharing a persistence length
-$p_u$ and per-domain contour lengths $L_{u1}$, the tension of the WLC
-is determine by summing the contour lengths
-\begin{equation}
- F(x, p_u, L_u, N_u) = F_\text{WLC}(x, p_u, N_uL_{u1})
-\end{equation}
-
-\subsection{Freely-jointed chains}
-\label{sec:tension:fjc}
+++ /dev/null
-\section{Double-integral Kramers' theory}
-
-The double-integral form of overdamped Kramers' theory may be too
-complex for analytical predictions of unfolding-force histograms.
-Rather than testing the entire \sawsim\ simulation (\cref{sec:sawsim}),
-we will focus on demonstrating that the Kramers' $k(F)$ evaluations
-are working properly. If the Bell modeled histograms check out, that
-gives reasonable support for the $k(F) \rightarrow \text{histogram}$
-portion of the simulation.
-
-Looking for analytic solutions to Kramers' $k(F)$, we find that there
-are not many available in a closed form. However, we do have analytic
-solutions for unforced $k$ for cusp-like and quartic potentials.
-
-\subsection{Cusp-like potentials}
-
-
-\subsection{Quartic potentials}
-
+++ /dev/null
-\section{Overview}
-
-For testing the \sawsim\ program, we need a few analytic solutions to unfolding distributions.
-We will start out discussing single-domain proteins under constant loading, and make some comments about multi-domain proteins and variable loading if we can make any progress in that direction.
-This note also functions as my mini-review article on unfolding theory, since
-I haven't been able to find an official one.
+++ /dev/null
-\section{Review of current research}
-
-\citet{rief02} provide a general review of force spectroscopy with a short section on protein unfolding.
-There's not all that much information here, but it's a good place to go to get
-a big-picture overview before diving into the more technical papers.
-
-There are two main approaches to modeling protein domain unfolding under tension: Bell's and Kramers'\citep{schlierf06,dudko06,hummer03}.
-Bell introduced his model in the context of cell adhesion\citep{bell78}, but it has been widely used to model mechanical unfolding in proteins\citep{rief97a,carrion-vazquez99b,schlierf06} due to it's simplicity and ease of use\citep{hummer03}.
-Kramers introduced his theory in the context of thermally activated barrier crossings, which is how we use it here.
-
-There is an excellent review of Kramers' theory in \citet{hanggi90}.
-The bell model is generally considered too elementary to be worth a detailed review in this context, and yet I had trouble finding explicit probability densities that matched my own in Eqn.~\ref{eq:unfold:bell_pdf}.
-Properties of the Bell model recieve more coverage under the name of the older and equivalent Gompertz distribution\citep{gompertz25,olshansky97,wu04}.
-A warning about the ``Gompertz'' model is in order, because there seem to be at least two unfolding/dying rate formulas that go by that name.
-Compare, for example, \citet{braverman08} Eqn.~5 and \citet{juckett93} Fig.~2.
-
-\subsection{Who's who}
-
-The field of mechanical protein unfolding is developing along three main branches.
-Some groups are predominantly theoretical,
-\begin{itemize}
- \item Evans, University of British Columbia (Emeritus) \\
- \url{http://www.physics.ubc.ca/php/directory/research/fac-1p.phtml?entnum=55}
- \item Thirumalai, University of Maryland \\
- \url{http://www.marylandbiophysics.umd.edu/}
- \item Onuchic, University of California, San Diego \\
- \url{http://guara.ucsd.edu/}
- \item Hyeon, Chung-Ang University (Onuchic postdoc, Thirumalai postdoc?) \\
- \url{http://physics.chem.cau.ac.kr/} \\
- \item Dietz (Rief grad) \\
- \url{http://www.hd-web.de/}
- \item Hummer and Szabo, National Institute of Diabetes and Digestive and Kidney Diseases \\
- \url{http://intramural.niddk.nih.gov/research/faculty.asp?People_ID=1615}
- \url{http://intramural.niddk.nih.gov/research/faculty.asp?People_ID=1559}
-\end{itemize}
-and the experimentalists are usually either AFM based
-\begin{itemize}
- \item Rief, Technischen Universität München \\
- \url{http://cell.e22.physik.tu-muenchen.de/gruppematthias/index.html}
- \item Fernandez, Columbia University \\
- \url{http://www.columbia.edu/cu/biology/faculty/fernandez/FernandezLabWebsite/}
- \item Oberhauser, University of Texas Medical Branch (Fernandez postdoc) \\
- \url{http://www.utmb.edu/ncb/Faculty/OberhauserAndres.html}
- \item Marszalek, Duke University (Fernandez postdoc) \\
- \url{http://smfs.pratt.duke.edu/homepage/lab.htm}
- \item Guoliang Yang, Drexel University \\
- \url{http://www.physics.drexel.edu/~gyang/}
- \item Wojcikiewicz, University of Miami \\
- \url{http://chroma.med.miami.edu/physiol/faculty-wojcikiewicz_e.htm}
-\end{itemize}
-or laser-tweezers based
-\begin{itemize}
- \item Bustamante, University of California, Berkley \\
- \url{http://alice.berkeley.edu/}
- \item Forde, Simon Fraser University \\
- \url{http://www.sfu.ca/fordelab/index.html}
-\end{itemize}
-
-\subsection{Evolution of unfolding modeling}
-
-Evans introduced the saddle-point Kramers' approximation in a protein unfolding context 1997 (\citet{evans97} Eqn.~3).
-However, early work on mechanical unfolding focused on the simper Bell model\citep{rief97a}.%TODO
-In the early `00's, the saddle-point/steepest-descent approximation to Kramer's model (\citet{hanggi90} Eqn.~4.56c) was introduced into our field\citep{dudko03,hyeon03}.%TODO
-By the mid `00's, the full-blown double-integral form of Kramer's model (\citet{hanggi90} Eqn.~4.56b) was in use\citep{schlierf06}.%TODO
-
-There have been some tangential attempts towards even fancier models.
-\citet{dudko03} attempted to reduce the restrictions of the single-unfolding-path model.
-\citet{hyeon03} attempted to measure the local roughness using temperature dependent unfolding.
-
-\subsection{History of simulations}
-
-Early molecular dynamics (MD) work on receptor-ligand breakage by Grubmuller 1996 and Izrailev 1997 (according to Evans 1997).
-\citet{evans97} introduce a smart Monte Carlo (SMC) Kramers' simulation.
-
-\subsection{History of experimental AFM unfolding experiments}
-
-\begin{itemize}
- \item \citet{rief97a}:
-\end{itemize}
-
-\subsection{History of experimental laser tweezer unfolding experiments}
-
-\begin{itemize}
- \item \citet{izrailev97}:
-\end{itemize}
+++ /dev/null
-\section{Single-domain proteins under constant loading}
-
-Let $x$ be the end to end distance of the protein, $t$ be the time since loading began, $F$ be tension applied to the protein, $P$ be the surviving population of folded proteins.
-Make the definitions
-\begin{align}
- v &\equiv \deriv{t}{x} && \text{the pulling velocity} \\
- k &\equiv \deriv{x}{F} && \text{the loading spring constant} \\
- P_0 &\equiv P(t=0) && \text{the initial number of folded proteins} \\
- D &\equiv P_0 - P && \text{the number of dead (unfolded) proteins} \\
- \kappa &\equiv -\frac{1}{P} \deriv{t}{P} && \text{the unfolding rate}
-\end{align}
-\nomenclature{$\equiv$}{Defined as (\ie equivalent to)}
-The proteins are under constant loading because
-\begin{equation}
- \deriv{t}{F} = \deriv{x}{F}\deriv{t}{x} = kv\;,
-\end{equation}
-a constant, since both $k$ and $v$ are constant (\citet{evans97} in the text on the first page, \citet{dudko06} in the text just before Eqn.~4).
-
-The instantaneous likelyhood of a protein unfolding is given by $\deriv{F}{D}$, and the unfolding histogram is merely this function discretized over a bin of width $W$(This is similar to \citet{dudko06} Eqn.~2, remembering that $\dot{F}=kv$, that their probability density is not a histogram ($W=1$), and that their pdf is normalized to $N=1$).
-\begin{equation}
- h(F) \equiv \deriv{\text{bin}}{F}
- = \deriv{F}{D} \cdot \deriv{\text{bin}}{F}
- = W \deriv{F}{D}
- = -W \deriv{F}{P}
- = -W \deriv{t}{P} \deriv{F}{t}
- = \frac{W}{vk} P\kappa \label{eq:unfold:hist}
-\end{equation}
-Solving for theoretical histograms is merely a question of taking your chosen $\kappa$, solving for $P(f)$, and plugging into Eqn. \ref{eq:unfold:hist}.
-We can also make a bit of progress solving for $P$ in terms of $\kappa$ as follows:
-\begin{align}
- \kappa &\equiv -\frac{1}{P} \deriv{t}{P} \\
- -\kappa \dd t \cdot \deriv{t}{F} &= \frac{\dd P}{P} \\
- \frac{-1}{kv} \int \kappa \dd F &= \ln(P) + c \\
- P &= C\exp{\p({\frac{-1}{kv}\integral{}{}{F}{\kappa}})} \;, \label{eq:P}
-\end{align}
-where $c \equiv \ln(C)$ is a constant of integration scaling $P$.
-
-\subsection{Constant unfolding rate}
-
-In the extremely weak tension regime, the proteins' unfolding rate is independent of tension, we have
-\begin{align}
- P &= C\exp{\p({\frac{-1}{kv}\integral{}{}{F}{\kappa}})}
- = C\exp{\p({\frac{-1}{kv}\kappa F})}
- = C\exp{\p({\frac{-\kappa F}{kv}})} \\
- P(0) &\equiv P_0 = C\exp(0) = C \\
- h(F) &= \frac{W}{vk} P \kappa
- = \frac{W\kappa P_0}{vk} \exp{\p({\frac{-\kappa F}{kv}})}
-\end{align}
-So, a constant unfolding-rate/hazard-function gives exponential decay.
-Not the most earth shattering result, but it's a comforting first step, and it does show explicitly the dependence in terms of the various unfolding-specific parameters.
-
-\subsection{Bell model}
-
-Stepping up the intensity a bit, we come to Bell's model for unfolding
-(\citet{hummer03} Eqn.~1 and the first paragraph of \citet{dudko06} and \citet{dudko07}).
-\begin{equation}
- \kappa = \kappa_0 \cdot \exp\p({\frac{F \dd x}{k_B T}})
- = \kappa_0 \cdot \exp(a F) \;,
-\end{equation}
-where we've defined $a \equiv \dd x/k_B T$ to bundle some constants together.
-The unfolding histogram is then given by
-\begin{align}
- P &= C\exp\p({\frac{-1}{kv}\integral{}{}{F}{\kappa}})
- = C\exp\p[{\frac{-1}{kv} \frac{\kappa_0}{a} \exp(a F)}]
- = C\exp\p[{\frac{-\kappa_0}{akv}\exp(a F)}] \\
- P(0) &\equiv P_0 = C\exp\p({\frac{-\kappa_0}{akv}}) \\
- C &= P_0 \exp\p({\frac{\kappa_0}{akv}}) \\
- P &= P_0 \exp\p\{{\frac{\kappa_0}{akv}[1-\exp(a F)]}\} \\
- h(F) &= \frac{W}{vk} P \kappa
- = \frac{W}{vk} P_0 \exp\p\{{\frac{\kappa_0}{akv}[1-\exp(a F)]}\} \kappa_0 \exp(a F)
- = \frac{W\kappa_0 P_0}{vk} \exp\p\{{a F + \frac{\kappa_0}{akv}[1-\exp(a F)]}\} \label{eq:unfold:bell_pdf}\;.
-\end{align}
-The $F$ dependent behavior reduces to
-\begin{equation}
- h(F) \propto \exp\p[{a F - b\exp(a F)}] \;,
-\end{equation}
-where $b \equiv \kappa_0/akv \equiv \kappa_0 k_B T / k v \dd x$ is
-another constant rephrasing.
-
-This looks similar to the Gompertz / Gumbel / Fisher-Tippett
-distribution, where
-\begin{align}
- p(x) &\propto z\exp(-z) \\
- z &\equiv \exp\p({-\frac{x-\mu}{\beta}}) \;,
-\end{align}
-but we have
-\begin{equation}
- p(x) \propto z\exp(-bz) \;.
-\end{equation}
-Strangely, the Gumbel distribution is supposed to derive from an
-exponentially increasing hazard function, which is where we started
-for our derivation. I haven't been able to find a good explaination
-of this discrepancy yet, but I have found a source that echos my
-result (\citet{wu04} Eqn.~1). TODO: compare \citet{wu04} with
-my successful derivation in \cref{sec:sawsim:results-scaffold}.
-
-Oh wait, we can do this:
-\begin{equation}
- p(x) \propto z\exp(-bz) = \frac{1}{b} z'\exp(-z')\propto z'\exp(-z') \;,
-\end{equation}
-with $z'\equiv bz$. I feel silly... From
-\href{Wolfram}{http://mathworld.wolfram.com/GumbelDistribution.html},
-the mean of the Gumbel probability density
-\begin{equation}
- P(x) = \frac{1}{\beta} \exp\p[{\frac{x-\alpha}{\beta}
- -\exp\p({\frac{x-\alpha}{\beta}})
- }]
-\end{equation}
-is given by $\mu=\alpha-\gamma\beta$, and the variance is
-$\sigma^2=\frac{1}{6}\pi^2\beta^2$, where $\gamma=0.57721566\ldots$ is
-the Euler-Mascheroni constant. Selecting $\beta=1/a=k_BT/\dd x$,
-$\alpha=-\beta\ln(\kappa\beta/kv)$, and $F=x$ we have
-\begin{align}
- P(F)
- &= \frac{1}{\beta} \exp\p[{\frac{F+\beta\ln(\kappa\beta/kv)}{\beta}
- -\exp\p({\frac{F+\beta\ln(\kappa\beta/kv)}
- {\beta}})
- }] \\
- &= \frac{1}{\beta} \exp(F/\beta)\exp[\ln(\kappa\beta/kv)]
- \exp\p\{{-\exp(F/\beta)\exp[\ln(\kappa\beta/kv)]}\} \\
- &= \frac{1}{\beta} \frac{\kappa\beta}{kv} \exp(F/\beta)
- \exp\p[{-\kappa\beta/kv\exp(F/\beta)}] \\
- &= \frac{\kappa}{kv} \exp(F/\beta)\exp[-\kappa\beta/kv\exp(F/\beta)] \\
- &= \frac{\kappa}{kv} \exp(F/\beta - \kappa\beta/kv\exp(F/\beta)] \\
- &= \frac{\kappa}{kv} \exp(aF - \kappa/akv\exp(aF)] \\
- &= \frac{\kappa}{kv} \exp(aF - b\exp(aF)]
- \propto h(F) \;.
-\end{align}
-So our unfolding force histogram for a single Bell domain under
-constant loading does indeed follow the Gumbel distribution.
-
-% Consolidate with src/sawsim/discussion.tex
-
-\subsection{Saddle-point Kramers' model}
-
-For the saddle-point approximation for Kramers' model for unfolding
-(\citet{evans97} Eqn.~3, \citet{hanggi90} Eqn. 4.56c, \citet{vanKampen07} Eqn. XIII.2.2).
-\begin{equation}
- \kappa = \frac{D}{l_b l_{ts}} \cdot \exp\p({\frac{-E_b(F)}{k_B T}}) \;,
-\end{equation}
-where $E_b(F)$ is the barrier height under an external force $F$,
-$D$ is the diffusion constant of the protein conformation along the reaction coordinate,
-$l_b$ is the characteristic length of the bound state $l_b \equiv 1/\rho_b$,
-$\rho_b$ is the density of states in the bound state, and
-$l_{ts}$ is the characteristic length of the transition state
-\begin{equation}
- l_{ts} = TODO
-\end{equation}
-
-\citet{evans97} solved this unfolding rate for both inverse power law potentials and cusp potentials.
-
-\subsubsection{Inverse power law potentials}
-
-\begin{equation}
- E(x) = \frac{-A}{x^n}
-\end{equation}
-(e.g. $n=6$ for a van der Waals interaction, see \citet{evans97} in
-the text on page 1544, in the first paragraph of the section
-\emph{Dissociation under force from an inverse power law attraction}).
-Evans then goes into diffusion constants that depend on the
-protein's end to end distance, and I haven't worked out the math
-yet. TODO: clean up.
-
-
-\subsubsection{Cusp potentials}
-
-\begin{equation}
- E(x) = \frac{1}{2}\kappa_a \p({\frac{x}{x_a}})^2
-\end{equation}
-(see \citet{evans97} in the text on page 1545, in the first paragraph
-of the section \emph{Dissociation under force from a deep harmonic well}).
+++ /dev/null
-\section{Theoretical unfolding force distributions}
-\label{sec:unfolding-distributions}
-
-\input{unfolding/distributions-overview}
-\input{unfolding/distributions-review}
-\input{unfolding/distributions-single_domain-constant_loading}
-\input{unfolding/distributions-kramers}
+++ /dev/null
-\chapter{Unfolding Theory}
-\label{sec:unfolding}
-
-\input{unfolding/rate}
-\input{unfolding/tension}
-\input{unfolding/distributions}
+++ /dev/null
-\subsection{Bell model}
-\label{sec:rate:bell}
+++ /dev/null
-\subsection{Kramers' saddle-point approximation}
-\label{sec:rate:kramers-saddle}
+++ /dev/null
-\subsection{Kramers' double integral model}
-\label{sec:rate:kramers}
+++ /dev/null
-\subsection{Stiffness-corrected Bell model}
-\label{sec:rate:stiff-bell}
+++ /dev/null
-\section{Single-domain unfolding rates}
-\label{sec:rate}
-
-\input{unfolding/rate-overview}
-\input{unfolding/rate-bell}
-\input{unfolding/rate-stiff-bell}
-\input{unfolding/rate-kramers}
-\input{unfolding/rate-kramers-saddle}
+++ /dev/null
-\subsection{Freely-jointed chains}
-\label{sec:tension:fjc}
+++ /dev/null
-\subsection{Folded domain tension}
-\label{sec:tension:folded}
-
-The unfolded polypeptide chain has been shown to follow the
-WLC\index{WLC} model quite well (\cref{sec:tension:wlc}), though other
-polymer models, such as the Freely-Jointed Chain
-(FJC)\citep{verdier70}\index{FJC}\nomenclature{FJC}{Freely-Jointed Chain}
-(\cref{sec:tension:fjc}), can be used to fit the force-extension
-relationship\citep{janshoff00}. A short chain of folded proteins,
-however, cannot be described well by polymer models. Several studies
-have used WLC and FJC models to fit the elastic properties of the
-modular protein titin\citep{granzier97,linke98a},
-% TODO: check it really is folded domains \& bulk titin
-but native titin contains hundreds of folded and unfolded domains.
-For the short protein polymers common in mechanical unfolding
-experiments, the cantilever dominates the elasticity of the
-polymer-cantilever system before any protein molecules unfold. After
-the first unfolding event occurs, the unfolded portion of the chain is
-already longer and softer than the sum of all the remaining folded
-domains, and dominates the elasticity of the whole chain. Therefore,
-the details of the tension model chosen for the folded domains has
-negligible effect on the unfolding forces, which was also suggested by
-\citet{staple08}. Force curves simulated using different models to
-describe the folded domains yielded almost identical unfolding force
-distributions (data not shown, TODO: show data).
+++ /dev/null
-\subsection{Wormlike chains}
-\label{sec:tension:wlc}
-
-The unfolded forms of many domains can be modeled as Worm-Like Chains
-(WLCs)\citep{marko95,bustamante94}
-\index{WLC}\nomenclature{WLC}{Wormlike Chain}, which treats the
-unfolded polymer as an elastic rod of persistence length $p$ and
-contour length $L$. The relationship between tension $F$ and
-extension (end-to-end distance) $x$ is given to within XX\% by
-Bustamante's interpolation formula\citep{marko95,bustamante94}.
-\begin{equation}
- F_\text{WLC}(x,p,L) = \frac{k_B T}{p_u}
- \p[{ \frac{1}{4}\p({ \frac{1}{(1-x/L)^2} - 1 })
- + \frac{x}{L} }] \;,
- \label{eq:sawsim:wlc}
-\end{equation}
-where $p$ is the persistence length.
-
-For chain with $N_u$ unfolded domains sharing a persistence length
-$p_u$ and per-domain contour lengths $L_{u1}$, the tension of the WLC
-is determine by summing the contour lengths
-\begin{equation}
- F(x, p_u, L_u, N_u) = F_\text{WLC}(x, p_u, N_uL_{u1})
-\end{equation}
+++ /dev/null
-\section{Chain Tension}
-\label{sec:tension}
-
-unfolded domains: polymer models (WLC, FJC)
-
-The particular model used for folded domains is of negligable
-importance in modeling chains with small numbers of domains, as
-discussed in \cref{sec:tension:folded}.
-
-\input{unfolding/tension-wlc}
-\input{unfolding/tension-fjc}
-\input{unfolding/tension-folded}
projects / thesis.git / commitdiff