Statistical Analysis of Lateral Diffusion and Reaction Kinetics of Single Molecules on the Membranes of Living Cells

Abstract

Single-molecule imaging has made it possible to directly observe the behavior of signaling molecules functioning on the membrane of living cells, revealing multiple subpopulations which can be characterized by their lateral diffusion coefficients on the membrane and or kinetics of dissociation from the membrane. The transition kinetics between these functional states is a central problem for understanding bio-signaling mechanisms. Here I propose a novel method to simultaneously analyze lateral diffusion coefficient and reaction kinetics from single-molecule trajectories. Based on the probability density function of displacement derived from a diffusion equation with appropriate reaction terms, the temporal development of diffusive mobility can be analyzed in a quantitative manner. I discuss simple diffusion models for a molecule that exhibits one or two states with different diffusion coefficients in the absence or presence of state transitions and/or membrane dissociation. I use numerical simulation based on my model to generate single-molecule trajectories to demonstrate the practice of this method with special emphasis on revealing reaction schemes.

Appendix

When calculated from experimental trajectories, the autocorrelation function of the squared displacements in model 6 contains an error term dependent on the lag time, τ. Let us assume the trajectory consists of an estimated molecular position at time t, x′(t), that distributes around the actual position, x(t), with fluctuations, η(t), such that the variance is ε ² and <η(t)> = 0. The displacement along the x-axis during time t and t + Δt, Δx ′(t), is written as,

$$ \Delta x\,'(t) = \Delta x(t) + \Delta \eta (t), $$

where

$$ \Delta x(t) = x\left( {t + \Delta t} \right) - x(t), $$

$$ \Delta \eta (t) = \eta \left( {t + \Delta t} \right) - \eta (t). $$

The variance of Δη(t) is 2ε ². Then, an autocorrelation function calculated from the trajectories of all molecules theoretically follows,

$$ \begin{array}{llll} \left\langle {\Delta {{x\,'}^2}(0)\Delta {{x'}^2}\left( \tau \right)} \right\rangle = \left\langle {\Delta {x^2}(0)\Delta {x^2}\left( \tau \right)} \right\rangle + \left\langle {\Delta {x^2}(0)} \right\rangle \left\langle {\Delta {\eta^2}\left( \tau \right)} \right\rangle \cr \quad+ 4\left\langle {\Delta x(0)\Delta x\left( \tau \right)} \right\rangle \left\langle {\Delta \eta (0)\Delta \eta \left( \tau \right)} \right\rangle + 2\left\langle {\Delta x(0)} \right\rangle \left\langle {\Delta \eta (0)\Delta {\eta^2}\left( \tau \right)} \right\rangle \cr \quad+ \left\langle {\Delta {x^2}\left( \tau \right)} \right\rangle \left\langle {\Delta {\eta^2}(0)} \right\rangle + 2\left\langle {\Delta x\left( \tau \right)} \right\rangle \left\langle {\Delta {\eta^2}(0)\Delta \eta \left( \tau \right)} \right\rangle \\ \quad+ \left\langle {\Delta {\eta^2}(0)\Delta {\eta^2}\left( \tau \right)} \right\rangle. \\\end{array} $$

When $\tau \ne 0$and $\tau \ne \Delta t$, it is,

$$ \begin{array}{llll} \left\langle {\Delta {{x\,'}^2}(0)\Delta {{x\,'}\,^2}\left( \tau \right)} \right\rangle = \left\langle {\Delta {x^2}(0)\Delta {x^2}\left( \tau \right)} \right\rangle + \left\langle {\Delta {x^2}(0)} \right\rangle \left\{ {\left\langle {{\eta^2}\left( {\tau + \Delta t} \right)} \right\rangle + \left\langle {{\eta^2}\left( \tau \right)} \right\rangle } \right\} \cr \quad+ \left\langle {\Delta {x^2}\left( \tau \right)} \right\rangle \left\{ {\left\langle {{\eta^2}\left( {\Delta t} \right)} \right\rangle + \left\langle {{\eta^2}(0)} \right\rangle } \right\} + \left\langle {{\eta^2}\left( {\Delta t} \right)} \right\rangle \left\langle {{\eta^2}\left( {\tau + \Delta t} \right)} \right\rangle \cr \quad+ \left\langle {{\eta^2}\left( {\Delta t} \right)} \right\rangle \left\langle {{\eta^2}\left( \tau \right)} \right\rangle + \left\langle {{\eta^2}(0)} \right\rangle \left\langle {{\eta^2}\left( {\tau + \Delta t} \right)} \right\rangle + \left\langle {{\eta^2}(0)} \right\rangle \left\langle {{\eta^2}\left( \tau \right)} \right\rangle, \\\end{array}$$

(12.14)

which is composed of three terms: an autocorrelation of the squared displacements in the absence of the error, <Δx ²(0)Δx ²(τ)>, an ensemble average of the error, <η ²(t)>, and an actual value of the ensemble-averaged squared displacements, <Δx ²(t)>. <Δx ²(0)Δx ²(τ)> is explained by Eq. 12.13.

If the molecule does not exhibit membrane dissociation, the ensemble average of the error is equal to ε ² irrespective of time t. However, in the presence of dissociation, the number of molecules decreases depending on t, leading to a concomitant decrease in the ensemble average. The ensemble average of the error imposed on molecules that dissociate at t=t _r is,

$$ \left\langle {{\eta^2}(t)|{t_r}} \right\rangle = {\varepsilon^2}H({t_r} - t) = \left\{ {\begin{array}{lll} {{\varepsilon^2},} & {t \leq {t_r},} \\{0,} & {t \geq {t_r},} \\\end{array} } \right. $$

where H(t) is a Heaviside function. Taking the integral for t _r from 0 to plus infinity, the ensemble average is,

$$ \begin{array}{lllll} \left\langle {{\eta^2}(t)} \right\rangle = \int_0^\infty {\left\langle {{\eta^2}(t)|{t_r}} \right\rangle } P\left( {{t_r}} \right)d{t_r} \\ = {\varepsilon^2}\int_0^\infty {H({t_r} - t)} P\left( {{t_r}} \right)d{t_r} \\ = {\varepsilon^2}\int_t^\infty {P\left( {{t_r}} \right)d{t_r}} \cr = {\varepsilon^2}\left( {1 - \int_0^t {P\left( {{t_r}} \right)d{t_r}} } \right) \\\end{array} $$

where P(t _r ) represents the probability that a molecule dissociates at t=t _r . Theoretically, the membrane residence probability, R(t), is equivalent to $1 - \int_0^t {P\left( {{t_r}} \right)d{t_r}}$. Experimentally, the release curve can be used as an estimate of R(t). Thus, the ensemble average of the error can be calculated from the experimental data.

The actual value of the ensemble-averaged squared displacements is calculated from the trajectories as follows. Since the estimated value is,

$$ \begin{array}{lll} \left\langle {\Delta {{x\,'}\,^2}(t)} \right\rangle = \left\langle {{{\left( {\Delta x(t) + \Delta \eta (t)} \right)}^2}} \right\rangle \\ = \left\langle {\Delta {x^2}(t)} \right\rangle + \left\langle {\Delta {\eta^2}(t)} \right\rangle \\ = \left\langle {\Delta {x^2}(t)} \right\rangle + \left\langle {{\eta^2}\left( {t + \Delta t} \right)} \right\rangle + \left\langle {{\eta^2}(t)} \right\rangle, \\\end{array} $$

the actual value of the ensemble-averaged squared displacements is,

$$ \left\langle {\Delta {x^2}(t)} \right\rangle = \left\langle {\Delta {{x\,'}^2}(t)} \right\rangle - \left\langle {{\eta^2}\left( {t + \Delta t} \right)} \right\rangle - \left\langle {{\eta^2}(t)} \right\rangle. $$

The ensemble average of the estimated squared displacements is obtained from the time series of squared displacements of i-th molecule,

$$ \left\langle {\Delta {{x\,'}^2}(t)} \right\rangle = \frac{1}{X}\sum\limits_{i = 1}^X {\Delta {{x\,'}_i}^2(t)}. $$

From Eq. 12.14 and calculating <η ²(t)> and <Δx ²(t)>, the autocorrelation function in the absence of the measurement error, <Δx ²(0)Δx ²(τ)>, is,

$$ \begin{array}{lll} \left\langle {\Delta {x^2}(0)\Delta {x^2}\left( \tau \right)} \right\rangle = \left\langle {\Delta {{x\,'}^2}(0)\Delta {{x\,'}^2}\left( \tau \right)} \right\rangle \cr \quad- {\varepsilon^2}\left[ {\left\langle {\Delta {{x\,'}^2}(0)} \right\rangle \left\{ {R\,'\left( {\tau + \Delta t} \right) + R'\left( \tau \right)} \right\} + \left\langle {\Delta {{x\,'}^2}\left( \tau \right)} \right\rangle \left\{ {R'\left( {\Delta t} \right) + 1} \right\}} \right] \\ \quad+ {\varepsilon^4}\left\{ {R'\left( {\Delta t} \right) + 1} \right\}\left\{ {R'\left( {\tau + \Delta t} \right) + R'\left( \tau \right)} \right\}, \\\end{array} $$

where R′(t) represent the release curve. The calculated autocorrelation function is fitted to Eq. 12.13 to obtain values of s ₁, s ₂, D and E.