From: W. Trevor King Date: Fri, 12 Mar 2010 04:23:19 +0000 (-0500) Subject: Fixed the bulk of Mom's remaining typos and clarification requests X-Git-Tag: v1.0~428 X-Git-Url: http://git.tremily.us/?a=commitdiff_plain;h=b9194147c0bda86ba00210e26858e9ed24bdcd40;p=thesis.git Fixed the bulk of Mom's remaining typos and clarification requests --- diff --git a/tex/src/cantilever-calib/solve_general.tex b/tex/src/cantilever-calib/solve_general.tex index 05d296d..261f497 100644 --- a/tex/src/cantilever-calib/solve_general.tex +++ b/tex/src/cantilever-calib/solve_general.tex @@ -40,7 +40,7 @@ Integrating over positive $\omega$ to find the total power per unit time yields &= \frac{G_0 \pi}{2m \beta k} \end{align} The integration is detailed in \cref{sec:integrals}. -By our corollary to Parseval's theorem (\cref{eq:parseval-var}), we have +By the corollary to Parseval's theorem (\cref{eq:parseval-var}), we have \begin{equation} \avg{x(t)^2} = \frac{G_0 \pi}{2m^2\beta\omega_0^2} \label{eq:DHO-var} \end{equation} diff --git a/tex/src/cantilever/methods.tex b/tex/src/cantilever/methods.tex index b60d1b5..5fbef6e 100644 --- a/tex/src/cantilever/methods.tex +++ b/tex/src/cantilever/methods.tex @@ -4,7 +4,7 @@ The experiments were carried out on octomers of I27 (\cref{fig:I27}). I27 is a model protein that has been used in mechanical unfolding experiments since the first use of synthetic -chains\citep{carrion-vazquez99b,TODO}. We use it here because it is +chains\citep{carrion-vazquez99b,TODO}. It was used here because it is both well characterized and readily available (% \href{http://www.athenaes.com/}{AthenaES}, Baltimore, MD, \href{http://www.athenaes.com/I27OAFMReferenceProtein.php}{0304}). @@ -23,7 +23,7 @@ the presense of the proposed intermediate\cite{TODO}. \label{fig:I27}} \end{figure} -The I27 octamers were stored in a ... buffer solution. +The I27 octamers were stored in a TODO buffer solution. Mechanical unfolding experiments were carried out on I27 octomers (AthenaES) in PBS on gold-coated coverslips. We used both cantilevers @@ -32,11 +32,11 @@ $20\U{pN/nm}$. Promising sawtooth curves were selected by eye and fit to WLCs\index{WLC} to identify I27 unfolding events. The results were sorted into two bins according to cantilever stiffness, and then averaged across each cantilever-stiffness/pulling-speed group to -produce the following graph. +produce \cref{fig:plot-splits}. \begin{figure} \includegraphics[width=4in]{figures/cantilever-data/plot_splits} - \caption{plot splits.} + \caption{plot splits.\label{fig:plot-splits}} \end{figure} Unfortunately, the data are not of high enough quality to extract the diff --git a/tex/src/cantilever/theory.tex b/tex/src/cantilever/theory.tex index e6ff413..0b07f3f 100644 --- a/tex/src/cantilever/theory.tex +++ b/tex/src/cantilever/theory.tex @@ -5,11 +5,11 @@ \caption{Energy landscape schematic.\label{fig:landscape}} \end{figure} -The presence of attached linkers and cantilever alters the free energy -landscape. Tension in the linkers favors domain unfolding, but that -tension is not necessarily independent of the unfolding reaction +The presence of attached linkers and cantilevers alters the free +energy landscape. Tension in the linkers favors domain unfolding, but +that tension is not necessarily independent of the unfolding reaction coordinate. For sufficiently stiff cantilevers and linkers, even the -small extension of the domain as it shifts from it's bound to +small extension of the domain as it shifts from its bound to transition state noticeably reduces the effective tension. Assuming the bound and transition state extensions are relatively independent of the applied tension, the energy of the transition state will be diff --git a/tex/src/introduction/main.tex b/tex/src/introduction/main.tex index 94c26e7..ef8c4b7 100644 --- a/tex/src/introduction/main.tex +++ b/tex/src/introduction/main.tex @@ -41,15 +41,17 @@ remarkably difficult. \begin{figure} \begin{center} \includegraphics[width=2in]{figures/biotin-streptavidin/1SWE.png}% - \caption{Complex of biotin (red) and a streptavidin tetramer (green) + \caption{Complex of biotin\index{biotin} (red) and a + streptavidin\index{streptavidin} tetramer (green) (\href{http://dx.doi.org/10.2210/pdb1swe/pdb}{PDB ID: 1SWE})% \citep{freitag97}. The correct streptavidin conformation creates - the biotin-specific binding pockets. Biotin-streptavidin is a model - ligand-receptor pair isolated from the bacterium - \species{Streptomyces avidinii}. Streptavidin binds to cell - surfaces, and bound biotin increases streptavidin's binding - affinity\citep{alon90}. - Figure generated with \citetalias{pymol}. + the biotin-specific binding pockets. Biotin-streptavidin is a + model ligand-receptor pair isolated from the bacterium + \species{Streptomyces avidinii}% + \index{Streptomyces@\species{Streptomyces avidnii}}. Streptavidin + binds to cell surfaces, and bound biotin increases streptavidin's + cell-binding affinity\citep{alon90}. Figure generated with + \citetalias{pymol}. \label{fig:ligand-receptor}} \end{center} \end{figure} @@ -87,14 +89,14 @@ explaining the folding mechanism. For a number of years, the \subfloat[][]{\includegraphics[width=2in]{figures/schematic/dill97-fig4}% \label{fig:folding:landscape}} \caption{(a) A ``double T'' example of the pathway model of protein - folding, in which the protein proceeds through a series of - metastable transition states $I_1$ and $I_2$ with two ``dead end'' - states $I_1^X$ and $I_2^X$. Adapted from \citet{bedard08}. (b) - The landscape model of protein folding, in which the protein - diffuses through a multi-dimensional free energy landscape. - Separate folding attempts may take many distinct routes through - this landscape on the way to the folded state. Reproduced from - \citet{dill97}. + folding, in which the protein proceeds from the native state $N$ + to the unfolded state $U$ via a series of metastable transition + states $I_1$ and $I_2$ with two ``dead end'' states $I_1^X$ and + $I_2^X$. Adapted from \citet{bedard08}. (b) The landscape model + of protein folding, in which the protein diffuses through a + multi-dimensional free energy landscape. Separate folding + attempts may take many distinct routes through this landscape on + the way to the folded state. Reproduced from \citet{dill97}. \label{fig:folding}} \end{center} \end{figure} @@ -146,22 +148,21 @@ stimulated much effort in both experimental and theoretical research. \section{Mechanical unfolding experiments} -% AFM unfolding procedure -In a mechanical unfolding experiment, a protein polymer is tethered -between two surfaces: a flat substrate and an AFM tip. The polymer is -stretched by increasing the separation between the two surfaces -(\cref{fig:unfolding-schematic}). The most common mode is the -constant speed experiment in which the substrate surface is moved away -from the tip at a uniform rate. The tethering surfaces, \ie, the AFM -tip and the substrate, have much larger radii of curvature than the -dimensions of single domain globular proteins that are normally used -for folding studies. This causes difficulties in manipulating -individual protein molecules because nonspecific interactions between -the AFM tip and the substrate may be stronger than the forces required -to unfold the protein when the surfaces are a few nanometers apart. -To circumvent these difficulties, globular protein molecules are -linked into polymers, which are then used in the AFM -studies\citep{carrion-vazquez99b,chyan04,carrion-vazquez03}. When +% AFM unfolding procedure In a mechanical unfolding experiment, a +protein polymer is tethered between two surfaces: a flat substrate and +an AFM tip. The polymer is stretched by increasing the separation +between the two surfaces (\cref{fig:unfolding-schematic}). The most +common mode is the constant speed experiment in which the substrate +surface is moved away from the tip at a uniform rate. The tethering +surfaces, \ie, the AFM tip and the substrate, have much larger radii +of curvature than the dimensions of single domain globular proteins +that are normally used for folding studies. This causes difficulties +in manipulating individual protein molecules because nonspecific +interactions between the AFM tip and the substrate may be stronger +than the forces required to unfold the protein when the surfaces are a +few nanometers apart. To circumvent these difficulties, globular +protein molecules are linked into polymers, which are then used in the +AFM studies\citep{carrion-vazquez99b,chyan04,carrion-vazquez03}. When such a polymer is pulled from its ends, each protein molecule feels the externally applied force, which increases the probability of unfolding by reducing the free energy barrier between the native and @@ -171,9 +172,10 @@ each protein molecule and prevents another unfolding event from occurring immediately. The force versus extension relationship, or \emph{force curve}, shows a typical sawtooth pattern (\cref{fig:expt-sawtooth}), where each peak corresponds to the -unfolding of a single protein in the polymer. Therefore, the +unfolding of a single protein domain in the polymer. Therefore, the individual unfolding events are separated from each other in space and -time, facilitating single molecule studies. +time, allowing single molecule resolution despite the use of +multi-domain test proteins. \begin{figure} \begin{center} @@ -211,11 +213,11 @@ tension modeling. \Cref{sec:unfolding-distributions} pulls mechanical unfolding experiments. This theory makes straightforward analysis of unfolding results difficult, so \cref{sec:sawsim} presents a Monte Carlo simulation approach to fitting unfolding parameters, and -\cref{sec:contour-space} presents the contour-length space approach to -fingerprinting unfolding pathways. \Cref{sec:temperature-theory} -wraps up the theory section by extending the analysis in -\cref{sec:unfolding,sec:unfolding-distributions} to multiple -temperatures. +\cref{sec:contour-space} presents the contour-length space analysis +for converting force curves to unfolding pathway fingerprints. +\Cref{sec:temperature-theory} wraps up the theory section by extending +the analysis in \cref{sec:unfolding,sec:unfolding-distributions} to +multiple temperatures. \Cref{sec:apparatus} describes our experimental apparatus and methods, as well as calibration procedures. With both the theory and procedure diff --git a/tex/src/packages.tex b/tex/src/packages.tex index d849278..de044d1 100644 --- a/tex/src/packages.tex +++ b/tex/src/packages.tex @@ -1,3 +1,7 @@ +% Remove the \leftmark (chapter title), since some chapter/section +% titles would overlap. +\fancyfoot[RE,LO]{} + \usepackage[super,sort&compress]{natbib} % fancy citation extensions % super selects citations in superscript mode % sort&compress automatically sorts and compresses compound citations (\cite{a,b,...}) diff --git a/tex/src/root.bib b/tex/src/root.bib index b7dc3be..13d7502 100644 --- a/tex/src/root.bib +++ b/tex/src/root.bib @@ -364,7 +364,7 @@ doi = "10.1038/sj.embor.7400403", URL = "http://www.nature.com/embor/journal/v6/n5/abs/7400403.html", eprint = "http://www.nature.com/embor/journal/v6/n5/pdf/7400403.pdf", - note = "Applies H&T\cite{hyeon03} to ligand-receptor + note = "Applies H\&T\cite{hyeon03} to ligand-receptor binding.", project = "Energy Landscape Roughness", } @@ -1435,7 +1435,7 @@ doi = "10.1073/pnas.120048697", URL = "http://www.pnas.org/cgi/content/abstract/97/12/6527", eprint = "http://www.pnas.org/cgi/reprint/97/12/6527.pdf", - note = "Unfolding order not from protein-surface interactions. XOXOXO... experiment.", + note = "Unfolding order not from protein-surface interactions. Mechanical unfolding of a chain of interleaved domains $ABABAB\ldots$ yielded a run of $A$ unfoldings followed by a run of $B$ unfoldings.", } @Article{nome07, @@ -2822,7 +2822,7 @@ eprint = {http://www.biophysj.org/cgi/reprint/72/4/1541.pdf}, } @Article{hanggi90, - title = {Reaction-rate theory: fifty years after Kramers}, + title = {Reaction-rate theory: fifty years after {K}ramers}, author = {H\"anggi, Peter and Talkner, Peter and Borkovec, Michal }, journal = {Rev. Mod. Phys.}, volume = {62}, @@ -3448,7 +3448,7 @@ eprint = {http://www.biophysj.org/cgi/reprint/93/10/3373.pdf} @Article{olshansky97, author = "S. J. Olshansky and B. A. Carnes", - title = "Ever since Gompertz", + title = "Ever since {G}ompertz", journal = "Demography", year = "1997", month = feb, @@ -3493,14 +3493,14 @@ url = "http://www.jstor.org/stable/2061656", @Article{juckett93, author = "D. A. Juckett and B. Rosenberg", - title = "Comparison of the Gompertz and Weibull functions as + title = "Comparison of the {G}ompertz and {W}eibull functions as descriptors for human mortality distributions and their intersections", journal = "Mech Ageing Dev", - year = "1993", + year = 1993, month = jun, - volume = "69", - number = "1-2", + volume = 69, + number = "1--2", pages = "1--31", keywords = "Adolescent", keywords = "Adult", @@ -5335,7 +5335,7 @@ doi = {10.1063/1.439715} @Article{walton08, author = "Emily B. Walton and Sunyoung Lee and Krystyn J. {Van Vliet}", - title = "Extending Bell's model: how force transducer stiffness + title = "Extending {B}ell's model: how force transducer stiffness alters measured unbinding forces and kinetics of molecular complexes", journal = BPJ, @@ -6524,9 +6524,8 @@ note = "I haven't read this, but it looks like a nice review of MD with constrai url = "http://stacks.iop.org/0953-8984/8/7561", eprint = "http://www.iop.org/EJ/article/0953-8984/8/41/006/c64103.pdf", project = "Cantilever Calibration", - note = "Actually write down Lagrangian formula and give a decent - derivation of PSD. Don't show or work out the integrals - though...", + note = "They actually write down Lagrangian formula and give a decent + derivation of PSD, but don't show or work out the integrals.", } @Article{staple08, diff --git a/tex/src/sawsim/discussion.tex b/tex/src/sawsim/discussion.tex index fc0519c..b5d458c 100644 --- a/tex/src/sawsim/discussion.tex +++ b/tex/src/sawsim/discussion.tex @@ -33,7 +33,7 @@ deviation of $25\U{pN}$. 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 point on a protein molecule is + 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}. @@ -131,7 +131,7 @@ 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 molecules experiments. These +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 @@ -156,7 +156,7 @@ unfolding order and polymer length. \includegraphics{figures/order-dep/fig} \caption{The dependence of the unfolding force on the temporal unfolding order for four polymers with $4$, $8$, $12$, and $16$ - molecules of identical proteins. Each point in the figure is the + identical protein domains. Each point in the figure is the average of $400$ data points. The first point in each curve represents the average of only the first peak in each of the $400$ simulated force curves, the second point represents the average of @@ -165,12 +165,11 @@ unfolding order and polymer length. $\kappa_\text{WLC}=203$, $207$, $161$, and $157\U{pN/nm}$, respectively, for lengths $4$ through $16$. The insets show the force distributions of the first, fourth, and eighth peaks, left - to right, for the polymer with eight protein molecules. The + to right, for the polymer with eight protein domains. The parameters used for generating the data were the same as those - used for \cref{fig:sawsim:sim-sawtooth}, except the polymer - length, and the histograms in the insets were normalized in the - same way as in - \cref{fig:sawsim:sim-hist}.\label{fig:sawsim:order-dep}} + used for \cref{fig:sawsim:sim-sawtooth}, except for the number of + domains. The histograms in the insets were normalized in the same + way as in \cref{fig:sawsim:sim-hist}.\label{fig:sawsim:order-dep}} \end{center} \end{figure} @@ -187,7 +186,7 @@ and persistence lengths of the protein polymer, the contour length increase from unfolding, and the stiffness (force constant) of the cantilever. Among these, the effect of the cantilever force constant is particularly interesting because cantilevers with a wide range of -force constants are available. In addition different single molecule +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 diff --git a/tex/src/sawsim/introduction.tex b/tex/src/sawsim/introduction.tex index c66afba..2652dbb 100644 --- a/tex/src/sawsim/introduction.tex +++ b/tex/src/sawsim/introduction.tex @@ -13,26 +13,26 @@ and energetic changes during force-induced protein unfolding. However, these simulations often involve time scales that are orders of magnitude smaller than those of the experiments, and the parameters used in the calculations are often neither experimentally controllable -nor measurable. As a result, a Monte Carlo simulation approach based -on a simple two-state kinetic model for the protein is usually used to -analyze data from mechanical unfolding experiments. A comparison of -the force curves measured experimentally and those generated from -simulations can yield the unfolding rate constant of the protein in -the absence of force as well as the distance from the native state to -the transition state along the pulling direction. The Monte Carlo -simulation method has been used since the first report of mechanical -unfolding experiment using -AFM\citep{rief97a,rief97b,rief98,carrion-vazquez99b,best02,zinober02,jollymore09}, -however, a comprehensive description and discussion of the simulation -procedures and the intricacies involved has not been reported. In -this paper, we provide a detailed description of the simulation -procedure, including the theories, approximations, and assumptions -involved. We also explain the procedure for extracting kinetic -properties of the protein from experimental data and introduce a -quantitative measure of fit quality between simulation and -experimental results. In addition, the effects of various -experimental parameters on force curve appearance are demonstrated, -and the errors associated with different methods of data pooling are -discussed. We believe that these results will be useful in -experimental design, artifact identification, and data analysis for -single molecule mechanical unfolding experiments. +nor measurable (TODO: example parameters of each type). As a result, +a Monte Carlo simulation approach based on a simple two-state kinetic +model for the protein is usually used to analyze data from mechanical +unfolding experiments. A comparison of the force curves measured +experimentally and those generated from simulations can yield the +unfolding rate constant of the protein in the absence of force as well +as the distance from the native state to the transition state along +the pulling direction. The Monte Carlo simulation method has been +used since the first report of mechanical unfolding experiments using +the AFM% +\citep{rief97a,rief97b,rief98,carrion-vazquez99b,best02,zinober02,jollymore09}, +however, a comprehensive discussion of the simulation procedures and +the intricacies involved has not been reported. In this paper, we +provide a detailed description of the simulation procedure, including +the theories, approximations, and assumptions involved. We also +explain the procedure for extracting kinetic properties of the protein +from experimental data and introduce a quantitative measure of fit +quality between simulation and experimental results. In addition, the +effects of various experimental parameters on force curve appearance +are demonstrated, and the errors associated with different methods of +data pooling are discussed. We believe that these results will be +useful in future experimental design, artifact identification, and +data analysis for single molecule mechanical unfolding experiments. diff --git a/tex/src/sawsim/methods.tex b/tex/src/sawsim/methods.tex index a28eac3..b4fa418 100644 --- a/tex/src/sawsim/methods.tex +++ b/tex/src/sawsim/methods.tex @@ -3,7 +3,7 @@ % simulation overview In simulating the mechanical unfolding process, a force curve is -generated by calculating the amount of the cantilever bending as the +generated by calculating the amount of cantilever bending as the substrate surface moves away from the tip. The cantilever bending is obtained by balancing the tension in the protein polymer and the Hookean force of the bent cantilever. The unfolding probability of @@ -39,7 +39,7 @@ respectively (\cref{fig:unfolding-schematic}). From this $F(x_t)$ may be computed using any multi-dimensional root-finding algorithm. % introduce particular models, and mention parameter aggregation -Inside this framework, we choose a particular extension model +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 \begin{equation} @@ -49,7 +49,7 @@ 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 +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} @@ -66,20 +66,20 @@ polymer-cantilever system is relatively insignificant. % address assumptions & caveats In the simulation, the protein polymer is assumed to be stretched in -the direction perpendicular to the surface, which is a good +the direction perpendicular to the substrate surface, which is a good approximation in most experimental situations, because the unfolded 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 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 +point and the position of the tip (\cref{eq:sawsim:x-total}). 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. % introduce constant velocity and walk through explicit example pull @@ -142,10 +142,10 @@ the cantilever deflection induced by liquid motion and fitting the time dependence of the deflection to an exponential function\cite{jones05}. For a $200\U{$\mu$m}$ rectangular cantilever with a bending spring constant of $20\U{pN/nm}$, the measured -relaxation time in water is $\sim50\U{$\mu$/s}$ (data not shown). -This relatively large relaxation time constant makes the cantilever -act as a low-pass filter and also causes a lag in the force -measurement. +relaxation time in water is $\sim50\U{$\mu$/s}$ (data not shown. +TODO: show data). This relatively large relaxation time constant +makes the cantilever act as a low-pass filter and also causes a lag in +the force measurement. \subsection{Unfolding protein molecules by force} \label{sec:sawsim:methods-unfolding} @@ -222,7 +222,7 @@ 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 -supered, because the unfolding transition is characterized by +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, diff --git a/tex/src/temperature-theory/main.tex b/tex/src/temperature-theory/main.tex index 1980f0e..e32d306 100644 --- a/tex/src/temperature-theory/main.tex +++ b/tex/src/temperature-theory/main.tex @@ -6,12 +6,14 @@ I'm skeptical about \HTeq{8} to \HTeq{9}, so I'll rework as much of their math as I am capable of\ldots -\begin{align} - \fs &= \frac{\kT}{\dx} \left[ \logp{ \frac{\r \dx}{\kexp \kT} } - + \logp{1 + \fs\frac{\dx'}{\dx} - \frac{\FO'}{\dx} + \frac{\vD'}{\vD}\cdot\frac{\kT}{\dx}} - + \logp{\avg{e^{\bt F_1}}}^2 \right] - & \HTeq{8} \nonumber -\end{align} +\begin{multline*} + \fs = \frac{\kT}{\dx} \Biggl[\Biggr. + \logp{ \frac{\r \dx}{\kexp \kT} } \\ + + \logp{1 + \fs\frac{\dx'}{\dx} - \frac{\FO'}{\dx} + \frac{\vD'}{\vD}\cdot\frac{\kT}{\dx}} \\ + + \logp{\avg{e^{\bt F_1}}}^2 + \Biggl.\Biggr] + \tag{\HTeq{8}} +\end{multline*} We simplify by dropping the 2\nd term (``In obtaining Eq.\ \textbf{9}, we have assumed that the second term in Eq.\ \textbf{8} is small.''), @@ -28,20 +30,23 @@ We obtain our version of \HTeq{9} by taking two measurements of equal mode force &= \frac{1}{\dx} \left( \alpha_1\rho_1 + \frac{\ep^2}{\alpha_1} -\alpha_2\rho_2 - \frac{\ep^2}{\alpha_2} \right) \\ \ep^2\left(\frac{1}{\alpha_2} - \frac{1}{\alpha_1}\right) &= \alpha_1\rho_1 - \alpha_2\rho_2 \\ - \ep^2 \cdot \frac{\alpha_1 - \alpha_2}{\alpha_1\alpha_2} &= \\ + \ep^2 \cdot \frac{\alpha_1 - \alpha_2}{\alpha_1\alpha_2} &= TODO\\ \ep^2 &= \frac{\alpha_1\alpha_2}{\alpha_1 - \alpha_2} \left( \alpha_1\rho_1 - \alpha_2\rho_2 \right)\\ - \ep^2 &= \frac{\kT_1\kT_2}{\kT_1 - \kT_2} \left[ \kT_1\logp{\frac{\rs1\dxs1}{\kexps1 \kT_1}} - - \kT_2\logp{\frac{\rs2\dxs2}{\kexps2 \kT_2}} \right] + \begin{split}\ep^2 &= \frac{\kT_1\kT_2}{\kT_1 - \kT_2} \Biggl[\Biggr. + \kT_1\logp{\frac{\rs1\dxs1}{\kexps1 \kT_1}} \\ + &\qquad- \kT_2\logp{\frac{\rs2\dxs2}{\kexps2 \kT_2}} + \Biggl.\Biggr] + \end{split} \end{align} Which is different from \HTeq{9} by the sign in the prefactor, and the replacement $\vD \rightarrow \kf$. -\begin{align} - \ep^2 &= \frac{\kT_1\kT_2}{\kT_2 - \kT_1} \left[ \kT_1\logp{\frac{\rs1\dxs1}{\vDs1 \kT_1}} - - \kT_2\logp{\frac{\rs2\dxs2}{\vDs2 \kT_2}} \right] - & \HTeq{9} \nonumber -\end{align} +\begin{equation*} + \ep^2 = \frac{\kT_1\kT_2}{\kT_2 - \kT_1} \left[ \kT_1\logp{\frac{\rs1\dxs1}{\vDs1 \kT_1}} + - \kT_2\logp{\frac{\rs2\dxs2}{\vDs2 \kT_2}} \right] + \tag{\HTeq{9}} +\end{equation*} -Alternatively, noting that \dx can vary as a function of temperature, we follow Nevo et al.\ in keeping it in. +Alternatively, noting that \dx can vary as a function of temperature, we follow \citet{nevo05} in keeping it in. Using $\delta \equiv \dx$ \begin{align} 0 &= \fs_1 - \fs_2 \\ @@ -53,7 +58,8 @@ Using $\delta \equiv \dx$ &= \frac{\delta_2\alpha_1\rho_1 - \delta_1\alpha_2\rho_2}{\delta_1\delta_2} \\ \ep^2 &= \frac{\alpha_1\alpha_2}{\delta_1\alpha_1 - \delta_2\alpha_2} \left( \delta_2\alpha_1\rho_1 - \delta_1\alpha_2\rho_2 \right)\\ - \ep^2 &= \frac{\kT_1\kT_2}{\dxs1\kT_1 - \dxs2\kT_2} - \left[ \dxs2\kT_1\logp{\frac{\rs1\dxs1}{\kfs1 \kT_1}} - - \dxs1\kT_2\logp{\frac{\rs2\dxs2}{\kfs2 \kT_2}} \right] + \begin{split}\ep^2 &= \frac{\kT_1\kT_2}{\dxs1\kT_1 - \dxs2\kT_2} + \Biggl[\Biggr. \dxs2\kT_1\logp{\frac{\rs1\dxs1}{\kfs1 \kT_1}} \\ + &\qquad - \dxs1\kT_2\logp{\frac{\rs2\dxs2}{\kfs2 \kT_2}} \Biggl.\Biggr] + \end{split} \end{align} diff --git a/tex/src/tension/folded.tex b/tex/src/tension/folded.tex index a2bebbc..3f77940 100644 --- a/tex/src/tension/folded.tex +++ b/tex/src/tension/folded.tex @@ -11,9 +11,9 @@ 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 -domains. For the short protein polymers common in mechanical -unfolding experiments, the cantilever dominates the elasticity of the +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 @@ -22,4 +22,4 @@ 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). +distributions (data not shown, TODO: show data). diff --git a/tex/src/unfolding-distributions/kramers.tex b/tex/src/unfolding-distributions/kramers.tex index 1d3efda..55ae7bf 100644 --- a/tex/src/unfolding-distributions/kramers.tex +++ b/tex/src/unfolding-distributions/kramers.tex @@ -9,9 +9,8 @@ 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 floating around in a finished form. However, we do have -analytic solutions for unforced $k$ for cusp-like and quartic -potentials. +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} diff --git a/tex/src/unfolding-distributions/review.tex b/tex/src/unfolding-distributions/review.tex index aa8b601..e0dec6e 100644 --- a/tex/src/unfolding-distributions/review.tex +++ b/tex/src/unfolding-distributions/review.tex @@ -6,7 +6,7 @@ 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{rief97b,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 crossing, which is how we use it here. +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}. @@ -58,19 +58,19 @@ or laser-tweezers based \subsection{Evolution of unfolding modeling} -Evans introduced? the saddle-point Kramers' approximation in a protein unfolding context 1997 (\citet{evans97} Eqn.~3). -Early work on mechanical unfolding focused on \citep{rief97b}.%TODO +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{rief97b}.%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 has been some tangential attempts towards even fancier models. +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} introduces a smart Monte Carlo (SMC) Kramers' simulation. +\citet{evans97} introduce a smart Monte Carlo (SMC) Kramers' simulation. \subsection{History of experimental AFM unfolding experiments} diff --git a/tex/src/unfolding-distributions/singledomain_constantloading.tex b/tex/src/unfolding-distributions/singledomain_constantloading.tex index bff4d5e..3dfa376 100644 --- a/tex/src/unfolding-distributions/singledomain_constantloading.tex +++ b/tex/src/unfolding-distributions/singledomain_constantloading.tex @@ -45,7 +45,7 @@ In the extremely weak tension regime, the proteins' unfolding rate is independen h(F) &= \frac{W}{vk} P \kappa = \frac{W\kappa P_0}{vk} \exp{\p({\frac{-\kappa F}{kv}})} \end{align} -Suprise! A constant unfolding-rate/hazard-function gives exponential decay. +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} @@ -76,7 +76,7 @@ The $F$ dependent behavior reduces to where $b \equiv \kappa_0/akv \equiv \kappa_0 k_B T / k v \dd x$ is another constant rephrasing. -This looks an awful lot like the the Gompertz/Gumbel/Fisher-Tippett +This looks similar to the Gompertz / Gumbel / Fisher-Tippett distribution, where \begin{align} p(x) &\propto z\exp(-z) \\ @@ -90,7 +90,8 @@ 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). +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} @@ -153,9 +154,9 @@ $l_{ts}$ is the characteristic length of the transition state (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 gets funky with diffusion constants that depend on the +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\ldots +yet. TODO: clean up. \subsubsection{Cusp potentials}