Biased random walk by stochastic fluctuations of chemoattractant-receptor interactions at the lower limit of detection

Abstract

Binding of ligand to its receptor is a stochastic process that exhibits fluctuations in time and space. In chemotaxis, this leads to a noisy input signal. Therefore, in a gradient of chemoattractant, the cell may occasionally experience a "wrong" gradient of occupied receptors. We obtained a simple equation for P(pos), the probability that half of the cell closest to the source of chemoattractant has higher receptor occupancy than the opposite half of the cell. P(pos) depends on four factors, the gradient property delC/sq. root of C, the receptor characteristic R(t)/K(D), a time-averaging constant I, and nonreceptor noise sigma(B). We measured chemotaxis of Dictyostelium cells to known shallow gradients of cAMP and obtained direct estimates for these constants. Furthermore, we observed that in shallow gradients, the measured chemotaxis index is correlated with P(pos), which suggests that chemotaxis in shallow gradients is a pure biased random walk. From the observed chemotaxis and derived time-averaging constant, we deduce that the gradient transducing second messenger has a lifetime of 2-8 s and a diffusion rate constant of approximately 1 microm(2)/s. Potential candidates for such second messengers are discussed.

FIGURE 1

Chemotactic activity of Dictyostelium cells toward cAMP. (A) Cells were stimulated with micropipettes containing different concentrations of cAMP as indicated. (B) Chemotaxis with the small population assay at different distances between cells and cAMP as indicated, or with the bridge assay, a modified Zigmond chamber. The results shown are the means of at least three independent experiments.

FIGURE 2

Measured chemotaxis index, and calculated P_pos at different distances from a micropipette containing 10⁻⁴ M cAMP. P_pos is the probability that the cell senses a gradient toward the pipette, and was calculated for different values of nonreceptor noise σ_B, sampling fold I, and receptor property R_t/K_D. (a) σ_B = 100, I = 1, R_t/K_D = 400; (b) σ_B =10, I = 1, R_t/K_D = 400; (c) σ_B = 0, I = 1, R_t/K_D = 400; (d) σ_B = 0, I = 3, R_t/K_D = 400; or σ_B = 0, I = 1, R_t/K_D = 1200; (e) σ_B = 0, I = 10, R_t/K_D = 400; or σ_B = 0, I = 1, R_t/K_D = 4000. The measured chemotaxis index (▴) presents the means and standard deviations of at least three independent experiments. (Inset) P_pos calculated for different values of SNR.

FIGURE 3

Plot of versus C to derive estimates for nonreceptor noise σ_B and sampling fold I. The chemotaxis data of Fig. 1 and Table 1, and the corresponding values of the cAMP concentration C and spatial gradient ∇C were used. (Solid symbols) Chemotaxis data of Table 1 at Ψ = 0.2. (Open symbols) All measured chemotaxis data of Fig. 1 at Ψ < 0.5. Linear regression of all data yields y = 0.0345x + 0.10; R² = 0.7423. The slope is and yields I = 1.82 (range 1.24–2.40; 95% confidence limits); the intercept with the ordinate is and yields σ_B = 25.2 (range 15.6–34.8). The dotted line is the linear regression for all data with σ_B = 0, yielding y = 0.0516x, R² = 0.387.

FIGURE 4

Chemotaxis index as function of P_pos. The chemotaxis index presented in Fig. 1 was determined with three assays using different experimental conditions. Equations A1–A6 were used to calculate the cAMP gradient around the cell in these assays, and Eqs. A8 and A9 were used to calculate P_pos, the probability that the cell senses a gradient toward the source of cAMP. The data are shown for P_pos < 0.95. In shallow gradients, when P_pos is small, the chemotaxis index is directly proportional to P_pos; in steeper gradients, chemotaxis reaches a maximum of Ψ = 0.8 when P_pos approaches 1. The dotted line is Ψ = P_pos(1–0.2 P_pos).

FIGURE 5

Dispersion length of diffusing second messengers as function of their lifetime and diffusion rate constant. The dispersion length of several potential second messengers was calculated with the equation where D_M is the diffusion rate constants (D, in μm²/s), τ_M is the lifetime of the second messenger, and γ is between 1 and 2 depending on the dimension of diffusion (γ = 1, 1.5, and 2 for receptor, lipid, and cGMP, respectively). The shaded areas indicate the response time of a cell to extend a pseudopod in the direction of a new gradient (∼3–8 s (–29,31,32)), and the 1–3 μm size of a pseudopod, respectively. The intersection of these shaded areas suggests that the transducing second messenger has a diffusion rate constant of 0.2–2 μm²/s.