Fluctuation correlation models for receptor immobilization

Nanoscale dynamics with cycles of receptor diffusion and immobilization by cell-external-or-internal factors is a key process in living cell adhesion phenomena at the origin of a plethora of signal transduction pathways. Motivated by modern correlation microscopy approaches, the receptor correlation functions in physical models based on diffusion-influenced reaction is studied. Using analytical and stochastic modeling, this paper focuses on the hybrid regime where diffusion and reaction are not truly separable. The time receptor autocorrelation functions are shown to be indexed by different time scales and their asymptotic expansions are given. Stochastic simulations show that this analysis can be extended to situations with a small number of molecules. It is also demonstrated that this analysis applies when receptor immobilization is coupled to environmental noise.

Figure 1

Receptors cycling between two conformational states diffuse on a two-dimensional membrane and they interact with immobilized ligand. Illumination through a Gaussian point-spread function (PSF) allows us to count the total number of receptors in the PSF region and to evaluate their time correlation functions.

Figure 2

Diagram for the receptor correlation function $I\left(t\right)$ for different sets of parameters. The two rate constants, $k_b^{★}$ and $k_u,b$ for binding and $u$ for unbinding, are renormalized by the diffusion time. The slanted doted line corresponds to the boundary above which reaction is faster than diffusion. The three functional forms $I_0\left(t\right),I_1\left(t\right)$, and its asymptotic form $I_{1,a}\left(t\right)$, are defined in the text; see Eqs. (21), (24), and (25). The effective diffusion limit matches the free diffusion for large unbinding rates $k_u$ as indicated by the arrows.

Figure 3

Snapshot of a stochastic simulation. This plot is a color density plot corresponding to ligated receptors $\mathrm{I}-\mathrm{L}$ in each compartment. The central disk corresponds to the adhesive region where the ligands are immobilized. Outside this region, the receptors freely diffuse without interacting with ligands and can exchange with a reservoir at the boundary of the square windowpane. The region $S$ of width $w$ for computing the correlation functions is inside the adhesive disk. The mean density of freely diffusing receptors inside and outside the disk are the same because of equal diffusion constants.

Figure 4

Plot of $C_{+}\left(q\right)$ (continuous lines) and $C_{-}\left(q\right)$ (dotted lines) for large and small fraction of bound receptors, $k_u/(k_u+k_b^{★})=0.99$ and 0.5. Functions are steplike in the limit of small bound fraction with a critical wave vector $q_i^{★}$; see Eq. (18).

Figure 5

This figure illustrates the Lax and Mergert argument [16] in the pure diffusion limit where the correlation $I\left(t\right)/I\left(0\right)$ is interpreted as the conditional probability for a random walker leaving the illuminated region of width $w$ at time $t=0$ to be observed in this region at a time $t$ later. The diffusion length is ${\left(Dt\right)}^{1/2}$. When $t$ is small, $I\left(t\right)/I\left(0\right)$ probes only random walkers at the boundary of the illuminated region. In the circular case where the boundary is much more sharply defined than in the Gaussian point-spread function case, one expects correlations to behave differently in the small time limit.

Figure 6

Plots of the autocorrelation functions in different regimes according to Fig. 2. In all cases, the dotted line corresponds to the numerical evaluation of $I\left(t\right)/I\left(0\right)$ versus $t/{\tau }_d,{\tau }_d=1$, using Eq. (15). In case (a), $k_b{\tau }_D=0.1,k_u{\tau }_D=0.5$ and in case (b), $k_b=0.01,k_u=0.01$. These two limit cases correspond to the regime where $I\left(t\right)$ is well approximated by the asymptotic form $I_{1,a}\left(t\right)$; cf. Eq. (25). The apparent plateau in the long time limit of case (b) is due to the small value of $k_u$. The dashed line gives the slope at the origin and the dotted line for the apparent value of the plateau is the fractional population of receptors in the bound state $k_b/(k_u+k_b)=0.5$.

Figure 7

Plots of the autocorrelation functions in different regimes according to Fig. 2. Case (a) corresponds to the dotted diagonal of Fig. 2 with $k_{+}{\tau }_D=k_{-}{\tau }_D=0.5$. The continuous line is $I_0\left(t\right)$ of Eq. (21). In case (b), $k_b{\tau }_D=k_u{\tau }_D=0.2$, and we have applied Eq. (24) for $I_1\left(t\right)$ (continuous line).

Figure 8

Schematic of the compartment-based model used in the stochastic simulation. The system of size $L$ is divided into $L/h\times L/h$ compartments playing the role of pixels. Diffusing receptors (red dots) can jump from one compartment to its neighbor with a rate $D/h^2$, where $D$ is the continuous space diffusion constant. Each compartment is treated as an homogenous system where interaction between molecules occurs according to first-order kinetics. Crosses schematize fixed ligand molecules, the spatial distribution of which is to choose to mimic the contact area. In the simulation presented Fig. 3, 20 ligand molecules are distributed in the compartments located near the center to mimic a circular illuminated volume. Compartments located at the boundaries of system $x,y=-10,+10$ are periodically reshuffled with a multinomial distribution to mimic an infinite reservoir.

Figure 9

Plot of the autocorrelation functions in the free diffusion limit. The two sets of data are renormalized by Eq. (28) and collapse on the universal $1/(1+x)$ curve $(w^2=16,52$ compartments). Numerical corrections are only perceptible at very short time where the numerical tangent is vertical.

Figure 10

Plots of the autocorrelation functions for different values of ligands and receptor densities. From the top to the bottom curve L = 30, 20, 10, with an average of $20$ receptors per compartment $\left(k_b={10}^{-3},k_u=5.0{10}^{-5}\right)$. The thin lines correspond to the mean-field result using effective rate constants as explained in the text; see Eqs. (30) and (31) with Eq. (25). Data correspond to reflective boundary conditions.

Figure 11

Plot of the receptor correlation function $\langle \delta n\left(t\right)\delta n\left(0\right)/\delta n{\left(0\right)}^2\rangle$ in the hybrid regime of part I. These plots evidence the presence of two time scales in the small and long time limit. The short time limit is characterized by the effective diffusion time defined in Eq. (19). The long time limit exhibits, however, an apparent plateau when $k_u$ is sufficiently small. From the top to the bottom, $k_u=5.0{10}^{-a},a=7,6,5,4$, with $k_b={10}^{-3}$. For these simulations, the average number of ligands per cell equal the average number of receptors per cell, i.e., 20 (reflective boundary conditions).

Figure 12

Figure illustrating the birth and death process by which immobile ligands with symbol X are created with rate $c$ and destroyed with rate $d$ in the adhesive disk; see Eq. (32). $\mathrm{I}-\mathrm{L}$ complexes are snapped with same probability.

Figure 13

Number of receptor-ligand complexes $\mathrm{I}-\mathrm{L}$ in a circular domain of size $w^2=52$ compartments. The time $t=0$ coincides with initial conditions with zero number of ligand-receptor complexes. The three cases shown correspond to slow (top curve) and fast ligand turnover. After a transient period the number of ligand-receptor pairs fluctuate around a mean and the mean decreases with increasing this rate. (a) $5.0{10}^{-3}{\text{s}}^{-1}$, (b) ${10}^{-2}{\text{s}}^{-1}$, (c) $5.0{10}^{-2}{\text{s}}^{-1}$.

Figure 14

Correlation function for the total number of receptors [Eq. (27)] in a circular domain of size $w^2=52$ compartments. From the top to the bottom curve, the characteristic rate for the fluctuations of the number of ligands per compartment increases. When this rate is large enough, the system is equivalent to the pure diffusion case (continuous blue line at the bottom). For a slower rate, top curve, correlations regress according to Eq. (25) with a long time tail (top continuous line in green adjusted to the mean filed value $k_b^{★}=k_b\langle \mathrm{L}\rangle )$. The characteristic frequency at which the system crosses over to the pure diffusive regime corresponds to the frequency of the binding-unbinding cycle of a receptor with its ligand $(k_b^{★}=k_u=0.01{\text{s}}^{-1})$. Top curve (a) $5{10}^{-4}{\text{s}}^{-1}$, (b) $5.0{10}^{-3}{\text{s}}^{-1}$, (c) $7,5{10}^{-3}{\text{s}}^{-1}$, (e) $5.0{10}^{-2}{\text{s}}^{-1}$, the continuous line almost coinciding with curve (e) corresponds to the pure diffusion problem. The effective diffusion constants for activated and nonactivated integrins are $0.1/{\text{h}}^2$ and $0.05/{\text{h}}^2$, where h is the compartment size. The apparent slope at the origin is 1 as stated in the text. Data correspond to open boundary conditions.

Figure 15

Plot of the correlation function for a sharp circular illuminated domain as a function of $t/{\tau }_D$. The doted line is the result of a stochastic simulation and the continuous line is a fit using reflective boundaries and a sharp circular domain. The inset details the neighborhood of the origin where the tangent is parallel to the vertical axis for both curves $(w^2=3$ (52 pixels), $k_b^{★}=0.02$ and $k_u=0.001)$.