Simulating Mono-and Multi-Protein Phosphorylation within Nanoclusters

TL;DR

The paper develops a 2D diffusion–reaction framework to quantify mono- and multi-phosphorylation within membrane nanoclusters, introducing a dimensionless parameter based on the diffusion length $l_p=\sqrt{D/\mu_p}$ relative to the activation disk radius $R_a$ and a desorption length $l_d=\sqrt{D/\mu_d}$. It provides analytic expressions for mono-phosphorylation probabilities under immortal ($\mathcal{P}_1(A)$) and mortal ($\mathcal{P}_1(B)$) boundary conditions, and shows how loop effects and dynamic disorder renormalize effective rates, enabling either switch-like or graded responses. The multi-phosphorylation analysis compares sequential and random networks and demonstrates that phosphatases can sharpen thresholds, indicating environment- and crosstalk-dependent control of phosphorylation states. The work links nanoscale biophysical constraints to signaling outcomes, offering insights into molecular self-assembly, condensate formation, and mechanobiology in adhesive structures.

Abstract

Protein nanoclustering is a characteristic feature of their activated state and is essential for forming numerous subcellular structures. The formation of these nanoclusters is highly dependent on a series of post-translational modifications, such as mono-and multi-phosphorylation and dephosphorylation of residues. We theoretically simulate how a protein can be either mono-or multi-phosphorylated on several residues in functional nanoclusters, depending on effective biophysical parameters (diffusion, dwell time, etc.). Moving beyond a binary view of phosphorylation, this approach highlights the interplay between mono-and multi-phosphorylation, the cooperative effects generally associated with multi-phosphorylation networks, and stresses the role of phosphatases in transforming graded phosphorylation signals into almost switch-like responses. The results are discussed in light of experiments that probe the distribution of phospho-residues.

Figures (17)

• Figure 1: An activation disk is a region divided into elementary cells where a protein can diffuse with diffusion coefficient $D = D_1$ and be phosphorylated at a rate of $\mu_p$. Each cell can oscillate over time between two states: active ($\mu_p >0$), red (light gray) symbols, and inactive ($\mu_p=0$), green (dark gray) symbols. This description is suitable for stochastic calculations. For the analytical approach, the effective medium approach means averaging the phosphorylation rate over the entire activation disk (yellow disk). Dephosphorylation will be introduced in a similar way with a local dephosphorylation rate $\mu_{dp}$.
• Figure 2: Four elementary processes for a protein initially deposited in the activation disk (yellow, see Fig. \ref{['fig:fig1']}) with diffusion coefficient $D_1$ inside the disk and $D_2$ outside: (1) The protein is phosphorylated before exiting the disk; (2) The protein is phosphorylated after a round trip (loop effect); (3) The protein is desorbed from the membrane outside the disk. Event (4) with the outside dashed circle only concerns numerical simulations and can be set aside on first reading, see Appendix \ref{['Appendix0']}.
• Figure 3: Probability of a protein being phosphorylated once as a function of the parameter $l_p/R_a$, where $l_p/R_a \ll 1$ corresponds to the strong phosphorylation regime. The bottom curve corresponds to model A (immortal protein) with absorbing boundary conditions at the boundary $r=R_a$, see (\ref{['eq:proba_one_model_A']}). Points correspond to numerical data with different values of $D, \, \mu_p$ and $R_a$ to illustrate how different systems collapse on the same curve. The two upper curves correspond to mortal proteins (model B, see (\ref{['eq:proba_one_model_B']})) for two values of $R_a=3$ (upper curve, $l_d/R_a\simeq 1.89$) and $R_a=6$ (median curve, $l_d/R_a \simeq 0.94$). The smaller the radius $R_a$, the greater the effect of round trips (loop effect).
• Figure 4: Average time for a protein to be phosphorylated at a single site, normalized by diffusion time, see (\ref{['eq:first_moment_resu']}). As indicated by the points obtained numerically, this curve, once normalized, is a scaling curve as a function of $l_p/R_a$. The dotted curve corresponds to $<t_{phos}> = 1/\mu_p$, which would be the value of the mean phosphorylation time for a disk of infinite radius. Down triangles and circles correspond to different sizes $R_a$ and make the scaling property explicit. For $l_p/R_a >1$, the probability that a protein will be phosphorylated is negligible.
• Figure 5: Normalized histograms of the number of times a protein crossed the boundary of the adhesive disk before being phosphorylated. There are therefore only even values. The two plots correspond to $l_p/R_a=0.21$ (orange - light gray) and $0.42$ (green - dark gray). For small $l_p/R_a$, in the high phosphorylation rate regime, proteins are rapidly phosphorylated. Data corresponds to $\mu_d = 5 \times 10^{-4}, \, R_a=3$ so that $l_d/R_a=1.89$.