From 82b030a175349faf917aad111b04c3bd86de621f Mon Sep 17 00:00:00 2001 From: "W. Trevor King" Date: Thu, 14 Jan 2010 14:13:51 -0500 Subject: [PATCH] Merged viscocity notes --- tex/src/root.tex | 1 + tex/src/unfolding_distributions/Makefile | 3 + tex/src/viscocity/Makefile | 3 + tex/src/viscocity/viscocity.tex | 80 ++++++++++++++++++++++++ 4 files changed, 87 insertions(+) create mode 100644 tex/src/unfolding_distributions/Makefile create mode 100644 tex/src/viscocity/Makefile create mode 100644 tex/src/viscocity/viscocity.tex diff --git a/tex/src/root.tex b/tex/src/root.tex index 0247a09..c227491 100644 --- a/tex/src/root.tex +++ b/tex/src/root.tex @@ -53,6 +53,7 @@ R01-GM071793. \appendix \include{cantilever_calib/cantilever_calib} +\include{viscocity/viscocity} \begin{vita} % Ph.D. only. See manual for details. diff --git a/tex/src/unfolding_distributions/Makefile b/tex/src/unfolding_distributions/Makefile new file mode 100644 index 0000000..a1f4f4c --- /dev/null +++ b/tex/src/unfolding_distributions/Makefile @@ -0,0 +1,3 @@ +all : + +clean : diff --git a/tex/src/viscocity/Makefile b/tex/src/viscocity/Makefile new file mode 100644 index 0000000..a1f4f4c --- /dev/null +++ b/tex/src/viscocity/Makefile @@ -0,0 +1,3 @@ +all : + +clean : diff --git a/tex/src/viscocity/viscocity.tex b/tex/src/viscocity/viscocity.tex new file mode 100644 index 0000000..167846d --- /dev/null +++ b/tex/src/viscocity/viscocity.tex @@ -0,0 +1,80 @@ +\linenumbers +\chapter{Hydrodynamic effects in fast AFM single-molecule force measurements} + +\begin{center} +{\Large M\"uller notes} \\ +\end{center} + +We had some trouble with their notation, so I'll try and clear some things up... + +\begin{center} +\begin{tabular}{r|l|l} + M\"uller \Bstrut & Trevor & Meaning \\ + \hline + $z_{surface}$ \Tstrut & $z_{surface}$ & Distance from the surface to the equilibrium cantilever position (increases on pulling) \\ + $z_{cantilever}$ & $z_{cantilever}$ & Cantilever deflection from it's equilibrium position (downward deflection positive) \\ + $h$ & $h$ & $h = z_{surface} - z_{cantilever}$ the distance between the tip and surface \\ + $v_{tip}$ & $v_{tip,surface}$ & $v_{tip,surface} = dh/dt$, tip velocity relative to the surface \\ + & $v_{eq,surface}$ & $v_{eq,surface} = dz_{surface}/dt$, pulling speed \\ + & $v_{tip,eq}$ & $v_{tip,eq} = dz_{cantilever}/dt$, tip velocity relative to its equilibrium position \\ + $F_{measured}$ & $F_{measured}$ & Measured force deflecting the cantilever \\ + $F_{net}$ & $F_{protein}$ & Force applied to stretch the protein \\ + $F_d$ & $F_d$ & Drag force acting on the cantilever \\ + $\Delta F$ & $F_{d:tip,eq}$ & Drag force due to only to $v_{tip,eq}$ \\ + & $F_{d:eq,surface}$ & Drag force due to only to $v_{eq,surface}$, the drag on an untethered cantilever \\ + & $F_{meas,zeroed}$ & $ F_{meas,zeroed} = F_{measured} - F_{d:eq,surface}$, defined for zero force in the detached region + \end{tabular} +\end{center} + +M\"uller equations: +\begin{align} + F_d &= \frac{6 \pi \eta a_{eff}^2}{h + d_{eff}} \cdot v_{tip} \label{mul_Fd} \\ + h &= z_{surface} - z_{cantilever} \\ + v_{tip} &= \frac{dh}{dt} \\ + \Delta F &= F_d(v,h) - F_d(v_{tip}, h) \label{mul_delF} \\ + F_{net} &= F_{measured} + \Delta F \label{mul_Fnet} +\end{align} + +Trevor derivations: \\ +For Eqn. \ref{mul_delF}, we assume that all the fluid in the cell moves with the surface + (i.e., fluid flow does not depend on height above the surface). +So the drag force is proportional to the speed of the tip relative to the surface. +\begin{equation} + F_d = D(h) v_{tip,surface} +\end{equation} +Where $D(h)$ is some constant that can depend on $h$ (like $6 \pi \eta a_{eff}^2 / (h + d_{eff})$). +This is M\"uller Eqn \ref{mul_Fd}. +Substituting in $v_{tip,surface} = v_{eq,surface} - v_{tip,eq}$ we have +\begin{align} + F_d &= D(h) v_{eq,surface} - D(h)v_{tip,eq} = F_{d:eq,surface} - F_{d:tip,eq} \\ + F_{d:tip, eq} &= F_{d:eq,surface} - F_d +\end{align} +This is M\"uller Eqn \ref{mul_delF}. +The measured force deflecting the cantilever is then +\begin{align} + F_{measured} &= F_{protein} + F_d \\ + F_{protein} &= F_{measured} - F_d = F_{measured} - (F_{d:eq,surface} - F_{d:tip,eq}) \\ + &= F_{meas,zeroed}' + F_{d:tip,eq} = F_{meas,zeroed}' + D(h)v_{tip,eq} +\end{align} +This is M\"uller Eqn \ref{mul_Fnet}. + +The treatment assumes the drag force on a detached cantilever doesn't depend on distance (see dashed line in Figure 4b,c), which doesn't make sense because +\begin{equation} + F_{d:eq,surface} = D(h)v_{eq,surface} +\end{equation} +And $D(h)$ depends on $h$. Therefore, this treatment uses $F_{meas,zeroed}'$, not $F_{meas,zeroed}$, where +\begin{equation} + F_{meas,zeroed}' = F_{measured} - F_{d:eq,surface,h\approx300nm} + = F_{meas,zeroed} + [D(h) - D(300nm)]v_{eq,surface} +\end{equation} +What can we do about this? + +The correction from $F_{meas,zeroed}'$ (solid line in Figure 3a) to $F_{protein}$ (dashed line) comes from adding $F_{d:tip,eq}$, which is why $F_{protein} = F_{meas,zeroed}'$ when +\begin{align} + 0 = F_{d:tip,eq} \propto v_{tip,eq} = \frac{dz_{cantilever}}{dt} \propto \frac{dF_{meas,zeroed}}{dt}, \\ +\end{align} +why $F_{protein} < F_{meas,zeroed}'$ when the cantilever is rebounding ($v_{tip,eq} < 0$), and +why $F_{protein} > F_{meas,zeroed}'$ when the cantilever is loading the protein ($v_{tip,eq} > 0$). + +This last ($F_{protein} > F_{meas,zeroed}'$ on loading) is why the raw +measurement \emph{underestimates} the unfolding force. -- 2.26.2