{"paper_id":"33877f4e-15b1-42ff-ba1d-40b66ed7829f","body_text":"Bridging molecular to cellular scales for models of membrane receptor signaling. \n \nKelvin J. Peterson, Boris M. Slepchenko and Leslie M. Loew* \n \nR. D. Berlin Center for Cell Analysis and Modeling, University of Connecticut School of \nMedicine, Farmington, CT USA \n \n*Correspondence to les@uchc.edu \n \n \nABSTRACT. Biochemical interactions at membranes are the starting points for cell signaling \nnetworks. But bimolecular reaction kinetics are difficult to experimentally measure on 2-\ndimensional membranes and are usually measured in volumetric in vitro assays. Membrane \ntethering produces confinement and steric effects that will significantly impact binding rates in \nways that are not readily estimated from volumetric measurements. Also, there are situations \nwhen 2D reactions do not conform to simple kinetics. Here we show how highly coarse-grained \nmolecular simulations using the SpringSaLaD software can be used to estimate membrane-\ntethered rate constants from experimentally determined volumetric kinetics. The approach is \nvalidated using an analytical solution for dimerization of binding sites anchored via stiff linkers. \nThis approach can provide 2-dimensional bimolecular rate constants to parameterize cell-scale \nmodels of receptor-mediated signaling. We explore how factors such as molecular reach, steric \neffects, disordered domains, local concentration and diffusion affect the kinetics of binding. We \nfind that for reaction-limited cases, the key determinant in converting 3D to 2D rate constant is \nthe distance of the binding sites from the membrane. On the other hand, the mass action rate \nlaw may no longer be obeyed for diffusion-limited reaction on surfaces; the simulations reveal \nwhen this situation pertains. We then apply our approach to epidermal growth factor receptor \n(EGFR) mediated activation of the membrane-bound small GTPase Ras. The analysis reveals \nhow prior binding of Ras to the allosteric site of SOS, a guanine nucleotide exchange factor \n(GEF) that is recruited to EGFR, significantly accelerates its catalytic activity. \n \nSIGNIFICANCE STATEMENT. In cell signaling, the activation of a surface receptor leads to a \ncascade of intracellular biochemical events. Many of these occur near the inner plasma \nmembrane surface. However, accurate rate parameters for these initial steps in models of \nsignaling are rarely available because membrane-tethered reaction kinetics are difficult to \nexperimentally measure. Here, we use a highly coarse-grained molecular simulator to model the \nkinetics of reactions between binding sites that are tethered to a membrane. We can fit these \nsimulation outputs to 2-dimensional rate laws to obtain rate constants that can be used to build \ncomplex models of cell signaling. These rate constants can also be compared to understand the \nkey biophysical features controlling the kinetics of bimolecular membrane reactions. \n  \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted December 5, 2024. ; https://doi.org/10.1101/2024.12.04.626844doi: bioRxiv preprint \n\nIntroduction. \n \nThe cell membrane responds to and integrates electrical, mechanical and chemical signals from \nthe extracellular environment. For chemical signals, the initial step is binding of a ligand to an \nexternal binding site on a membrane receptor protein. This triggers a chain of events that \ntypically involves a change of state of the cytoplasmic receptor domain and subsequent \nrecruitment of adapter proteins, enzymes and/or cytoskeletal regulators to further evoke a cell \nbiological response. Mathematical modeling of signaling pathways is a powerful tool to \nsystematically organize the experimental knowledge we have about these complex systems and \nthen develop predictions, through simulations, to inspire new experiments (1, 2).  \n \nA challenge in developing cell signaling models is the acquisition of the appropriate \nexperimentally-grounded input parameters. Often, kinetic data is available from in vitro \nbiochemistry and this has served the mature field of metabolic modeling very well. However, \nrate parameters are less available and more difficult to measure for signaling pathways and \nnetworks. Among the key challenges is that many of the essential steps are associated with the \nplasma membrane, where multiple molecules are recruited before a messenger ultimately \ndiffuses to an intracellular target (e.g. the nucleus). It is experimentally difficult to measure \nreaction rates on membranes, so available data is commonly derived from volumetric \nmeasurements. While such quantitative data is useful, it can be challenging to translate rate \nparameters derived from 3D solution to the biophysically very different environment of a 2D \nmembrane. \n \nIndeed, the biophysics of membrane associated reactions has a long scientific history.  An early \nfocus of investigation, initiated with a classic paper by Adam and Delbruck (3), was the \ndifference between 2D and 3D diffusion limited reactions; they argued that the 2-step process of \nabsorbing a cytosolic molecule to the membrane and subsequent 2D search for an enzyme or \nbinding partner might offer a kinetic advantage over a fully 3D search. However, this has been \ndisputed using subsequent experimentally determined realistic parameters and biological \nscenarios (4). With regard to cell signaling, theoretical analyses of reactions at membranes \nhave been extended and elaborated to consider both diffusion limited and reaction limited \nbimolecular kinetics(1, 5).  \n \nThese earlier pioneering studies treated membrane reactions as strictly two dimensional surface \nevents. However, most biological membrane associated reactions actually occur in the \nimmediately adjacent cytosol, with interacting sites tethered to the membrane through lipid or \nprotein anchors. A well-known feature of anchoring bimolecular reactions to a membrane is the \neffect of locally increased effective concentration: compared to the same reaction by the same \nnumber of molecules within the cell volume, anchoring the reaction to the membrane generally \nincreases concentration by confining the reaction volume to a thin layer above the membrane \n(6). The thickness of that layer is often parametrized as h, sometimes called the “confinement \nlength” (7, 8). Roughly, h is related to the distance the binding sites can sample above the \nmembrane surface. The smaller h, the greater the effective concentrations of binding sites and \nthe faster the membrane associated bimolecular kinetics. In principle, h, can be determined by \nanalyzing detailed molecular dynamics simulations on the flexibility and motions of binding \ndomains tethered to the membrane (8, 9). Recently, the concept of molecular reach was \nintroduced as a more general framework for assessing how  molecular structure can influence \nthe steady state phosphorylated fraction of a membrane bound substrate interacting with a \ntethered kinase (10). The reach is defined as the distance of the kinase site from the membrane \nanchor and is directly related to h when the anchor diffuses freely in the membrane. However, \nwhen lateral diffusion is restricted (e.g. within large signaling clusters such as the immune \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted December 5, 2024. ; https://doi.org/10.1101/2024.12.04.626844doi: bioRxiv preprint \n\nsynapse), binding sites with longer reach may have an advantage in being able to find more \nbinding partners; for such diffusion-limited scenarios, this can outweigh the decrease in local \nconcentration associated with increased h (10). \n \nThus, it is clear from these foundational studies that the membrane environment and the \nstructural features of interacting membrane-bound molecules need to be considered in \nconverting a measured 3D on-rate to a 2D on-rate suitable for cell-scale continuum models \nbased on ordinary or partial differential equations (ODEs or PDEs). In this work, we show how \nthis can be done using simulations from SpringSaLaD (11) to derive 2D rate constants. \nSpringSaLaD uses a series of variously sized spherical sites linked together with stiff springs to \ncoarsely model the key structural features of macromolecules such as flexibility, excluded \nvolume and binding site localization. Each sphere within the molecule is given its own diffusion \ncoefficient and Brownian diffusion is simulated via a Langevin dynamics algorithm. The \nmolecule can be tethered to a surface, representing a membrane, via a specialized anchoring \nsphere that can be given a lateral diffusion coefficient; the rest of the molecule, including the \nspheres designated as binding sites, are free to explore the volume above the membrane within \ntheir reach. Naturally therefore (and particularly advantageous for the purpose of this work), \nvolumetric rate constants are assigned to bimolecular rate expressions even for membrane-\nbound molecules. For example, this feature was used to show how multivalent clustering is \nenhanced when one of the interacting molecules  is tethered to a membrane (12).  \n \nTo determine 2D rate constants, we fit the stochastic kinetics simulated with SpringSaLaD to a \ndeterministic mass-action rate law based on the corresponding surface densities of the binding \npartners. Using idealized structures, we validate this procedure against analytical solutions. We \nexplore how molecular structural features, membrane density and lateral diffusion affect the \nkinetics. Then, as a biologically relevant example, we apply this approach to recruitment of \nmultivalent binding partners to the epidermal growth factor receptor (EGFR), including the \nbinding of SOS, the G-protein exchange factor (GEF) for Ras. A better understanding of the \ncooperativity of SOS activation of Ras emerges from this analysis. \n \nResults \n \nVolume vs. surface bi\n nding kinetics for a simple dimerization reaction. \n \nFor bi\nmolecular reactions, SpringSaLaD determines the microscopic probability of 2 binding \nsites forming a bond as they diffuse within a reaction radius that is slightly larger than the sum of \ntheir physical radii. The input to the algorithm is simply the macroscopic volumetric on rate \nconstant (𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑣𝑣𝑜𝑜𝑣𝑣), units of µM-1s-1) and the diffusion coefficient of the individual spheres. Full \ndetails on the derivation of the reaction probability and a thorough validation of its accuracy can \nbe found in the original paper describing SpringSaLaD (11).  Importantly for the purposes of this \nwork, the rate of binding for sites that happen to be tethered to a membrane are still treated as \nvolumetric, because the spherical sites are located in the volume compartment even while they \nare constrained with links to the 2D membrane surface.  \n \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted December 5, 2024. ; https://doi.org/10.1101/2024.12.04.626844doi: bioRxiv preprint \n\nFigure 1 illustrates this for a simple dimerization reaction where the 2 yellow binding sites are \ntethered to the membrane anchor (gray sphere) by a 5nm link; in these simulations both the \nanchor and tether sites are given identical diffusion coefficients of 1µm2/s. We ran 100 \nSpringSaLaD simulations each with 40 dimerizing molecules (Fig. 1 only shows 2 for clarity). \nThe mean trajectories for these 100 runs are then fit to a deterministic mass action membrane \nbinding model in terms of surface densities using either COPASI or Virtual Cell (although an \nanalytical solution can be also fit for simple dimerization) (Step 2 in Fig. 1). The output binding \nrate constant, 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚) in units of µm2molecule-1s-1, can then be used to parameterize larger \ndeterministic or stochastic models with molecule numbers (>1000) or timescales (>10 s) that \nwould be too large for even highly coarse-grained molecular simulators like SpringSaLaD. Also, \nwe emphasize that binding rates are typically determined experimentally using in vitro \nvolumetric measurements; that SpringSaLaD uses volumetric rate constants as its inputs makes \nthe procedure in Fig. 1 especially appropriate and convenient. The results in Fig. 1 show that a \n2nm diameter binding site with an on rate for dimerization of 0.47 µM\n-1s-1 and tethered to a \nmembrane surface through a 5nm link, can be modeled as a 2D surface reaction with an on \nrate, 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚)\n, of 0.109 µm2molecule-1s-1. Using idealized models, we now explore how various \nstructural and biophysical parameters control 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚)and when mass action rate constants may \nnot be appropriate to describe membrane-associated binding kinetics. \n \nFigure 1. Workflow for finding membrane bimolecular binding rate constants, 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚), in \nterms of surface densities. Step 1 is to run 100 SpringSaLaD trajectories with the \ncoarse grained model of the binding partners. Illustrated are a pair of simple monomers \n(top) consisting of 2nm diameter spheres (yellow) tethered to a 1nm membrane anchor \nsphere (gray) with a 5nm link; both the anchor and binding spheres are assigned the \ndiffusion coefficient of 1 µm\n2/s. At the bottom, the product dimer is depicted. Step 2 \nconsists of fitting the average of 100 outputs from stochastic SpringSaLaD (SS) \nsimulations to a deterministic (ODE) non-spatial model of surface-bound dimerization. \n \n=\n0.47µM-1s-1\n0\n50\n100\n150\n200\n250\n300\n350\n400\n450\n500\n0 0.05 0.1 0.15\nSurface Density of Dimer (molecule/µm2)\nTime (s)\nFit                = 0.109 µm2molecule-1s-1\nMembrane Dimer (SS)\nMembrane Dimer (fit)\nStep 1: Run 100 dimerization \nSimulations in SpringSaLaD\nStep 2: Fit mean of the 100  \nSpringSaLaD trajectories to find \nin a deterministic dimerization model \nusing surface densities\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted December 5, 2024. ; https://doi.org/10.1101/2024.12.04.626844doi: bioRxiv preprint \n\n \nEffects of volumetric on-r ate, 2D diffusion rate, surface density and structural features on \ndimerization kinetics \n \nWe s\ntart by analyzing the case of a 5nm stiff tether between a 1nm diameter binding sphere and \nthe membrane. The diffusion coefficient for the binder sphere is set to 1µm2/s. Table 1 gives \nresults for all combinations of two volumetric on rate constants, two surface densities and two \nmembrane diffusion coefficients. All the dimerization rate laws are irreversible except for the last \n2 rows, where the off-rate constant is indicated. The 2D on-rate constant derived by fitting the \nSpringSaLaD simulation, 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚), is in the 5th column; for consistency, all these fits are \nperformed for kinetics at 80% completion. For comparison we also provide the 2D on-rate \nconstant, 𝑘𝑘𝑜𝑜𝑜𝑜\n(ℎ) =  𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑣𝑣𝑜𝑜𝑣𝑣)/ℎ, where ℎ is the linker length plus the radius of the binder sphere. If ℎ is \nin units of µm, dividing by a unit conversion factor of 602.2 converts 𝑘𝑘𝑜𝑜𝑜𝑜\n(ℎ) from units of µM-1s-1 to \nunits of µm2molecules-1 s-1; 𝑘𝑘𝑜𝑜𝑜𝑜\n(ℎ) is an equivalent 2D binding rate constant of freely diffusing \nmonomers confined within a thin layer with a height ℎ adjacent to the membrane. The last \ncolumn provides the ratio of the sum of the squared deviations to the sum of the squared \nSpringSaLaD mean values; this ratio provides a measure of the goodness of fit to the \nbimolecular mass action rate law, with anything less than ~10-3 representing a good fit.  \nExamples of the fits are shown in Figure 2. \n \nWe l\nooked at two anchor diffusion coefficients corresponding to that of a large transmembrane \nprotein domain (𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚= 0.01 µm2/s) and a lipid anchor (𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚=1 µm2/s). The first row of Table 1 \nconsiders a case where diffusion of the binder (𝐷𝐷𝑣𝑣𝑜𝑜𝑣𝑣=1 µm2/s) is 100 times that of the anchor site \nin the membrane. Because the linker is stiff, the anchor acts as a pivot and the binder rapidly \nmoves within a hemispherical shell to effectively create a reaction region with a thickness of \nslightly larger than the 1-nm diameter of the binder sphere. Because the region of spatial \noverlap of the two shells where the binding may occur is restricted, it might seem surprising that  \n𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚) is so closely approximated by 𝑘𝑘𝑜𝑜𝑜𝑜\n(ℎ). However, as shown in the Supporting Information, \nthe reaction probability is enhanced because of the effectively higher density of binding sites \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted December 5, 2024. ; https://doi.org/10.1101/2024.12.04.626844doi: bioRxiv preprint \n\nwithin this restricted region, which compensates for the smaller spatial overlap. Thus, the \ncalculations in the Supporting Information both explain and validate the SpringSaLaD results. \nThe second row of Table 1 corresponds to the case where the anchor and the binder have the \nsame fast diffusion. In this scenario, that 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚)≅ 𝑘𝑘𝑜𝑜𝑜𝑜\n(ℎ) is intuitive, because the effect of the \ntether in this case essentially reduces to confining the binders within the layer adjacent to the \nmembrane. The solutions in the Supporting Information assume that the molecular distributions \nare spatially uniform at any time, which pertains to the cases of the first two rows of Table 1 \nwhere 𝑘𝑘𝑜𝑜𝑜𝑜\n(ℎ)<𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚.  \n \nI\nt has been shown that the rates of membrane reactions may be susceptible to deviations from \na simple mass action rate law (13-17), which manifests themselves as significant changes of the \napparent rate constant with initial surface density. These changes need to be assessed before \nusing the rate constants we obtain by the procedure of Fig. 1 in large scale cell-level models. \nWe probed for this by decreasing the initial density by a factor of 100. The combinations of \n𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚and 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑣𝑣𝑜𝑜\n𝑣𝑣) in the third and fourth rows of T able 1 resulted in relatively small changes in \n𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚), indicating that the mass action rate law applies to these cases. We further tested this by \nincreasing the 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑣𝑣𝑜𝑜\n𝑣𝑣) by a factor of 100 in the lower half of Table 1. Clearly, for the case of 𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚= \n0.01 µm2/s, there is a strong dependence on surface density. Furthermore, this value of 𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚  is \nbelow the “well-mixed” limit given by 𝑘𝑘𝑜𝑜𝑜𝑜\n(ℎ). Thus, the combination of 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑣𝑣𝑜𝑜\n𝑣𝑣)= 1.0 µM-1s-1 and \n𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚= 0.01 µm2/s present cases where mass action rate laws would not apply. In general, it \nwould not be appropriate to use a mass action rate law, if the apparent 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚) is significantly \ngreater than 𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 . Of course, the appropriateness of the mass action rate law can also be \njudged by the goodness of fit when determining 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚) from the SpringSaLaD simulation (last \ncolumn of Table 1); as demonstrated in Fig. 2, the case of 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑣𝑣𝑜𝑜\n𝑣𝑣)= 1.0 µM-1s-1 and 𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚= 0.01 \nµm2/s is not fit very well to a mass-action dimerization rate. The simulation results are initially \nfaster and then ultimately slower than the best fit that assumes mass action kinetics. This is \nbecause the monomers whose binding sites are initially close to each other (effectively within \nthe “reach” of the tether) will react, but leave behind depletion zones where monomers are too \nfar away from potential binding partners (15, 16). These isolated monomers can be discerned \ntoward the end of Movie 1, which presents an example trajectory for this case.  \n \nFigure 2. Examples of SpringSaLaD simulation data and their fit to surface-confined mass \naction dimerization kinetics. The initial surface density of monomers in each case is 2500 \nmolecule/µm2. The diffusion coefficient of the anchor and the volumetric binding rate constant \nof the binding sites are indicated above each graph, corresponding respectively to rows 1, 6 \nand 5 of Table 1. Each of the SpringSaLaD simulations were run to 80% completion. \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted December 5, 2024. ; https://doi.org/10.1101/2024.12.04.626844doi: bioRxiv preprint \n\nConsistent with the idea of depletion zones, the last 2 rows of Table 1 show that when \nreversibility is introduced, the fitted on-rate constant increases (compare, respectively, to rows 5 \nand 7 of Table 1). This is because when dimerization is reversible, free monomers can reappear \nto fill in depletion zones, thereby countering the slow diffusion. While the fits to mass action \nkinetics are still poor, especially for the low density case, the respective mean steady-state \ndensities of dimers in the SpringSaLaD simulations, 570 and 5.7  molecules/µm2, are consistent \nwith the thermodynamic law of mass action (18). According to this law, the ratio of the squared \nmonomer volumetric concentration and the dimer volumetric concentration is determined at \nsteady state by the dissociation constant, 𝐾𝐾𝑑𝑑= 𝑘𝑘𝑜𝑜𝑜𝑜𝑜𝑜/𝑘𝑘𝑜𝑜𝑜𝑜\n(ℎ). \n \nTabl\ne 2 provides results for three computational experiments in which structural features of the \nSpringSaLaD molecules are varied. These are all for dimerization reactions where the maximum \ndistance between the membrane and the binding site is 4 times longer than in Table 1: ℎ = \n0.0205 µm (linker length of 20nm and binding site radius of 0.5nm). The 𝑘𝑘𝑜𝑜𝑜𝑜\n(ℎ) is therefore a \nfactor of 4 slower than that in Table 1 for the same 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑣𝑣𝑜𝑜𝑣𝑣) of 1µM-1s-1.  The SpringSaLaD \nsimulations were carried out with a slow membrane-anchor diffusion coefficient,  𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚= \n0.01µm2/s, to model a membrane receptor. The first row can be directly compared to the fifth \nrow of Table 1, where the only difference is the length of the linker. As would be expected from \nthe increase in h, the membrane on rate constant is decreased; however, importantly, this \nconstant is now closer to the diffusion rate and therefore deviations from mass action are \nsignificantly reduced. The second row in Table 2 shows results for a structure where additional \nspherical sites are introduced between the membrane anchor and the binding site to model the \nspace occupied by a cytosolic protein sequence; this steric effect results in a small decrease in \n𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚). This decrease is reversed when flexibility is introduced by allowing the spherical sites to \nbe pivot points in the third row of the Table; this is how disordered domains may be modeled in \nSpringSaLaD. Overall, for these idealized structures and simple dimerization, the fitted on-rate \nconstants in Table 2 are relatively close to 𝑘𝑘𝑜𝑜𝑜𝑜\n(ℎ).  \n \n \nApplication of the method to interaction of receptor-bound  SOS with Ras. \n \nTill now, we have employed idealized molecular structures to validate our method and to learn \nsome biophysical principles that control the on-rate constants of binding sites tethered to \nmembranes. We now illustrate the application of this approach to a biologically relevant \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted December 5, 2024. ; https://doi.org/10.1101/2024.12.04.626844doi: bioRxiv preprint \n\nexample, namely the interaction of the G-protein exchange factor (GEF) SOS with the lipid \nanchored small G-protein Ras (19).  \n \nSOS has 2 binding sites for Ras: an allosteric site and a catalytic site. When a Ras molecule \nbinds to the allosteric site it increases the GEF activity of the catalytic site (20). Additionally, \nbefore SOS binds to Ras it is first recruited to an active receptor tyrosine kinase (RTK) through \nan adapter protein; the adaptor binds to a proline rich motif (PRM) on SOS via a SH3 domain \nand to a phosphorylated tyrosine via a SH2 domain. One such RTK is the Epidermal Growth \nFactor Receptor (EGFR) and one such adaptor protein is Grb2 (21). Once SOS is bound to \nGrb2, it becomes membrane tethered and its interaction with Ras is facilitated (20). However \nthe complex mechanistic details are still emerging (22).  \n \n \nFigure 3. Membrane binding of Ras to the catalytic site of Receptor-bound SOS. Top: \ndirect binding. Bottom: after pre-binding of Ras to the allosteric site. The molecular \nstructures are approximated by using the SpringSaLaD 3D editing utility based on atomic \nstructures derived from AlphaFold2. The top left structure is an EGFR cytoplasmic domain \nanchored to the membrane (red kinase domain, followed by yellow disordered tail capped by a \nphosphotyrosine in green); the latter is linked to a cyan SH2 domain in Grb2; one of its magenta \nSH3 domains is linked to an olive PRM on the end of the disordered region of SOS; the violet \nSOS binding site for the catalytic domain of Ras is indicated with an asterisk (*). The bottom left \nstructure is identical, except that the pink allosteric site on SOS is bound to Ras. The Ras \nstructures are shown in the center with the yellow binding sites indicated by an asterisk. The \ninput rate constants for the SpringSaLaD simulations are shown at the top, corresponding to the \nvolumetric on rate for Ras binding to the catalytic site of SOS. The EGFR anchor diffusion \ncoefficient is 0.01µm\n2/s. All other site diffusion coefficients are 1.0µm2/s. For each condition, 20 \nEGFR-Grb2-SOS molecules react with 200 Ras molecules on a 250nmX250nm membrane \nsurface to generate 100 SpringSaLaD trajectories. Their means were fitted to a deterministic 2D \nrate law to derive  𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚), fixing 𝑘𝑘𝑜𝑜𝑜𝑜𝑜𝑜 at 4.0 s-1; results for the 2 conditions are shown on the  \nright. \n \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted December 5, 2024. ; https://doi.org/10.1101/2024.12.04.626844doi: bioRxiv preprint \n\nWe asked the limited question of how binding of Ras with the receptor-associated SOS catalytic \nsite might depend on whether SOS is prebound to Ras at the allosteric site. Ras is a lipid-\nanchored protein, so we reasoned that binding of Ras to the allosteric site of SOS would bring \nthe SOS catalytic site closer to the membrane to enhance binding to a second Ras and \nsubsequent exchange of GDP for GTP . Just how big an effect this is, may be estimated by the  \n \nprocedure developed above, with the results shown in Fig. 3.  \n \nWe dev\neloped molecular models with the aid of the mol2sphere (23) utility within SpringSaLaD \nand were guided by AlphaFold 2 atomic structure predictions (24, 25); all the site diameters and \nlinker lengths are available in the SpringSaLaD input file included in the supporting information; \nsnapshots of the structure are shown in Fig. 3. The top of Fig. 3 displays results for binding of \nSOS-Grb2-EGFR  to Ras at the SOS catalytic site; the bottom shows results for the same \nreaction, except SOS-Grb2-EGFR had been first bound to a Ras molecule at the SOS allosteric \nsite. The on and off rates shown at the top of Fig. 3 (19) are applied to both of the reactions. In \nthese models, the EGFR membrane anchor site is assigned a diffusion coefficient of 0.01 µm\n2/s \nto represent a large transmembrane protein, while the Ras membrane anchor is assigned a \ndiffusion coefficient of 1.0 µm\n2/s to represent a lipid anchor; all the sites that are dangling in the \ncytosol volume are given 𝐷𝐷𝑣𝑣𝑜𝑜 𝑣𝑣 of 1.0 µm2/s, but since the binding reaction is not close to \ndiffusion-limited, the precise values are not critical. Consistent with these being reaction-limited \non-rates, the SpringSaLaD simulation outputs (averages of 40 runs) are well fitted to reversible \nmass action kinetic law, as shown in the plots on the right of Fig. 3 (relative squared deviations \nare, respectively, 4.1 X 10-4 and 2.0 X 10-4). Importantly, the 2D on-rate constants (𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚)) \nderived from these fits are, respectively, 1.1 X 10-3 µm2molecules-1s-1 and 4.0 X 10-3 \nµm2molecules-1s-1. Likewise, the affinity of the catalytic is site is increased by allosteric site pre-\nassociation: 𝐾𝐾𝑑𝑑 = 3600 molecules/µm2 for the top of Fig. 3 and 1000 molecules/µm2 for the \nbottom pre-association case. Thus, SOS allosteric site association with Ras is estimated to \naccelerate its catalytic site binding and affinity by a factor of ~4 – even when SOS is already \nconfined to the membrane through Grb2-mediated association with EGFR.  \n \nD\niscussion \n \nThe k\ninetics of reactions at membranes have long fascinated biophysicists (1, 3-5, 8, 10, 14, \n17). These studies have produced theoretical insights to illuminate how surface-associated \nreactions have distinct properties compared with reactions occurring in 3D solution. Which of \nthese special properties are most pertinent to any given membrane-bound molecular interaction \nis difficult to ascertain a priori. Furthermore, experiments to measure bimolecular kinetics on \nmembrane surfaces are difficult, so often only on-rate constants measured in 3D are accessible. \nFundamentally, however, the kinetics of key membrane-associated reactions depend on the 2D \nsurface densities and 2D rate constants, not on the bulk cellular concentrations and 3D rate \nconstants. Indeed, because surface to volume ratios of different cell types vary tremendously, \nvolumetric rate constants cannot be readily used to model and simulate cell signaling systems.  \nTo address these theoretical and practical problems, we describe a procedure (Fig. 1) using \nexperimentally accessible volumetric on-rate constants, 𝑘𝑘\n𝑜𝑜𝑜𝑜\n(𝑣𝑣𝑜𝑜\n𝑣𝑣) with the SpringSaLaD simulation \nsoftware to estimate the 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚), the 2-dimensional rate constant for a membrane-confined \nbimolecular reaction.  \n \nT\no validate the method, we applied it to the dimerization of a single binding site tethered to a \nsurface through a 5nm stiff linker, where the membrane anchor acts as a pivot (Table 1). For the \nsituation where the reaction is rate limiting, this system can be solved analytically (see \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted December 5, 2024. ; https://doi.org/10.1101/2024.12.04.626844doi: bioRxiv preprint \n\nSupporting Information); gratifyingly, 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚) determined by our method is well reproduced by \nthe analytical solution. Interestingly, for these cases, 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚) is well approximated by 𝑘𝑘𝑜𝑜𝑜𝑜\n(ℎ) =\n 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑣𝑣𝑜𝑜\n𝑣𝑣)/ℎ/602.2 (µm2molecules-1s-1), where ℎ is the distance of the binding site from the \nmembrane anchor. This parameter has also been referred to as the “confinement length” (8), \ndefining a thin volume above the membrane that concentrates the binding sites and directly \nproducing the relationship between 𝑘𝑘𝑜𝑜𝑜𝑜\n(ℎ) and 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑣𝑣\n𝑜𝑜𝑣𝑣).  \n \nW\nhile the mass action kinetics are generally applicable for both encounter-limited and reaction-\nlimited kinetics in 3D solution (but see (15)), it has long been appreciated that the situation may \nbe more complex for 2D kinetics (3, 5, 14, 17, 26). This is demonstrated by the results in Table 1 \nfor situations where the diffusion coefficient of the anchor is slow, but the volumetric on-rate \nconstant is fast. For these cases, different estimates of 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚) are obtained at different initial \nsurface density – clearly incompatible with the mass action kinetics. Indeed, the third panel of \nFigure 2 shows that the SpringSaLaD kinetic data is not well fitted by a 2D mass action rate law. \nA video of one of these trajectories (Movie 1) nicely illustrates how the initial rate is fast, while \nthe binding sites are within “reach” (10), but falls off as binding sites are left orphaned outside \nthe reach of the remaining slowly diffusing monomers. The results allow us to generalize that \nmass action applies as long as 𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 is close to or greater than 𝑘𝑘𝑜𝑜𝑜𝑜\n(ℎ). Furthermore, a mass \naction rate law applies well to the initial rate, before depletion zones develop. Thus, our analysis \nprovides an approach to determine whether 2D binding might be well approximated by the mass \naction kinetics, and if so, to estimate the 2D mass action on-rate constant. \n \nTo explore how other molecular structural features might affect dimerization of the monomers \ntethered to the membrane, we looked at 3 additional idealized systems in Table 2. In all these, h \nwas 20.5nm (as opposed to 5.5nm in Table 1). As expected, the longer confinement length \ndecreased the estimated 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚) by a factor of about 4. The insertion of steric sites between the \nanchor and the binding site or allowing for flexibility of the linker region have minor effects on \n𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚), which is relatively well approximated by 𝑘𝑘𝑜𝑜𝑜𝑜\n(ℎ)\n. Importantly, the lower value of 𝑘𝑘𝑜𝑜𝑜𝑜\n(ℎ) for this \nsystem was closer to 𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚, resulting in a better fit by the mass action kinetics than for the \nsimilar case with the 5nm linker (5th row of Table 1).  \n \nFor membrane binding of real biological molecules, it may be difficult to estimate the average \nlocation of a binding sites relative to the membrane surface (i.e. h); also, the 2 binding sites may \nbe parts of very different structures with different distances from the membrane surface. In \nsituations like this, our approach has the potential to provide good estimates of rate constants \nthat can be applied to larger cell signaling systems. Indeed, there may be direct insights that \ncan be realized just by considering the structural details of the interacting membrane molecules. \nWe have illustrated this in relation to adaptor-mediated protein kinase receptor signaling \nmechanisms, specifically for the interaction of the GEF SOS with its effector Ras (Fig. 3). It has \nbeen shown that direct catalysis by the SOS catalytic domain of Ras conversion from the GDP \nto the GTP states is relatively slow. However, prebinding binding of Ras to SOS at a site that is \nnot catalytic (termed the “allosteric” site on SOS) significantly accelerates the catalytic activity, \nwhere the catalysis becomes processive (19, 20, 22). The results in Fig. 3 suggest that at least \npart of this acceleration may be due to the close proximity of the SOS catalytic site to the \nmembrane once it is bound to Ras at its allosteric site. Even though SOS is already localized to \nthe membrane by initially binding to EGFR via Grb2 in our computational experiment, pre-\nbinding of the SOS allosteric site to Ras brings it to still closer proximity to the membrane. Of \ncourse, there could be additional effects such as a direct allosteric enhancement through a \nconformational change or release of self-inhibition (19, 20, 22), but here we focus on the \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted December 5, 2024. ; https://doi.org/10.1101/2024.12.04.626844doi: bioRxiv preprint \n\nsignificance of constricting the binding zone through membrane tethers of varying length and \nflexibility. Our approach toward deriving membrane on-rates will aid in the parametrization of \nODE and PDE models that could help elucidate the full kinetic and mechanistic details of \nprocesses such as Ras activation and downstream signaling. \n \n \nMethods \n \nAll simulations were performed with SpringSaLaD v. 2.3.4 (https://vcell.org/ssalad). To build \ncoarse grained molecular structures used in SpringSaLaD for Fig. 3, we used AlphaFold2 (24, \n25) to generate PDB file estimates of protein structures for EGFR, Grb2, SOS, and Ras via \ninput of entire amino acid sequences. These PDB files are converted to coarse-grained \nmolecular models via the mol2sphere (23) utility embedded in SpringSaLaD. In some cases, we \nmanually edited the structures to capture their essential features from measurements on the \nPDB structures, as visualized in PyMol (Schrödinger, Inc.). In particular, for SOS we subdivide \nthe CDC25 and REM domains into multiple smaller spherical sites with only one site capable of \nparticipating in a binding reaction. This process maintains the structural characteristics of these \ndomains, while ensuring that the binding radius of the domain is not artificially inflated. We also \nconsider molecule flexibility when making user modifications to molecule structure. When PDB \nfiles are imported to SpringSaLaD via mol2sphere, the default is for each site to be linked to no \nmore than 2 immediately adjacent sites. Flexibility can be decreased by introducing more stiff \nlinkers to more adjacent spherical sites. We employ this method to ensure that our multi-site \nrepresentation of CDC25 and REM domains diffuse as a fixed group of spherical sites instead of \nindividual, highly flexible domains. Modeling disordered regions, such as the PRM region of \nSOS, can be challenging due to low confidence in the AlphaFold2-generated geometry of these \nregions. To model disordered domains, we use PyMOL to measure the length of entire straight \nchain amino acid sequences, then model this sequence in SpringSaLaD using 1.0 nm diameter \nsites connected by 3.1 nm linkers. Binding reactions in all simulations have rates input in terms \nof µM\n-1s-1; successful binding results in 1nm (Tables 1 and 2)  or  0.5 nm (Fig. 3) distances \nbetween the surfaces of the spherical sites. A set of 100 trajectories are simulated in parallel \nusing the Center for Cell Analysis and Modeling High Performance Compute Cluster \n(https://health.uconn.edu/high-per formance-computing/resources/). SpringSaLaD input files are \nin Supporting Information and provide all the geometric details for the molecules in each \ncomputational experiment. \n \nThe m\nean of 100 SpringSaLaD simulations for each of the simulations in Tables 1 and 2 and \nFig. 3 were fit to a deterministic 2D binding rate law to obtain  𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚). For irrreversible \ndimerization (Table 1, first 8 rows and Table 2), a fit to an analytical expression (Eq. 1) used the \nExcel solver; for reversible dimerization (Table 1, rows 9 and 10) and for the fits in Fig. 3, we \nused the COPASI (27) parameter estimation tool within Virtual Cell (VCell) (28, 29). The latter \ncan be accessed in the VCell published BioModel “Peterson Figure 3: Ras-\nSOS_Binding_fit_to_SpringSaLaD”. All these results with some further analysis can also be \nfound in the spreadsheets included in the Supporting Information. \n \n \nAcknowledgments \nThis work was supported by NIH grants R24 GM137787 and R01 GM132859. We are pleased \nto acknowledge the advice of Aniruddha Chattaraj with some of the data analysis. \n \n \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted December 5, 2024. ; https://doi.org/10.1101/2024.12.04.626844doi: bioRxiv preprint \n\n1. J. M. Haugh, A Unified Model for Signal Transduction Reactions in Cellular Membranes. \nBiophys. J. 82, 591-604 (2002). \n2. B. M. Slepchenko, J. C. Schaff, I. Macara, L. M. Loew, Quantitative cell biology with the \nVirtual Cell. Trends Cell Biol 13, 570-576 (2003). \n3. G. Adam, M. Delbrück, \"Reduction of Dimensionality in Biological Diffusion\" in Structural \nChemistry and Molecular Biology, A. Rich, N. Davidson, Eds. (W. H. Freeman and Co., \nSan Francisco, 1968), pp. 198-215. \n4. M. A. McCloskey, M. Poo, Rates of Membrane-associated Reactions: Reduction of \nDimentionality Revisited. Journal of Cell Biology 102, 88-96 (1986). \n5. D. Axelrod, M. D. Wang, Reduction-of-dimensionality kinetics at reaction-limited cell \nsurface receptors. Biophys J 66, 588-600 (1994). \n6. B. Windisch, D. Bray, T. Duke, Balls and Chains—A Mesoscopic Approach to Tethered \nProtein Domains. Biophysical Journal 91, 2383-2392 (2006). \n7. G. I. Bell, M. Dembo, P . Bongrand, Cell adhesion. Competition between nonspecific \nrepulsion and specific bonding. Biophys J 45, 1051-1064 (1984). \n8. Y . Wu, J. Vendome, L. Shapiro, A. Ben-Shaul, B. Honig, Transforming binding affinities \nfrom three dimensions to two with application to cadherin clustering. Nature 475, 510-\n513 (2011). \n9. Z.-R. Xie, J. Chen, Y . Wu, Linking 3D and 2D binding kinetics of membrane proteins by \nmultiscale simulations. Protein Science 23, 1789-1799 (2014). \n10. Y . Zhang et al., The Influence of Molecular Reach and Diffusivity on the Efficacy of \nMembrane-Confined Reactions. Biophys J 117, 1189-1201 (2019). \n11. P . J. Michalski, L. M. Loew, SpringSaLaD: A Spatial, Particle-Based Biochemical \nSimulation Platform with Excluded Volume. Biophys J 110, 523-529 (2016). \n12. A. Chattaraj, M. Youngstrom, L. M. Loew, The interplay of structural and cellular \nbiophysics controls clustering of multivalent molecules. bioRxiv (accepted in Biophys. J.) \n10.1101/373084, 373084 (2018). \n13. R. Kopelman, Rate processes on fractals: Theory, simulations, and experiments. Journal \nof Statistical Physics 42, 185-200 (1986). \n14. H. Berry, Monte Carlo Simulations of Enzyme Reactions in Two Dimensions: Fractal \nKinetics and Spatial Segregation. Biophysical Journal 83, 1891-1901 (2002). \n15. A. A. Ovchinnikov, Y . B. Zeldovich, Role of density fluctuations in bimolecular reaction \nkinetics. Chemical Physics 28, 215-218 (1977). \n16. D. Toussaint, F. Wilczek, Particle–antiparticle annihilation in diffusive motion. The Journal \nof Chemical Physics 78, 2642-2647 (1983). \n17. D. C. Torney, H. M. McConnell, G. R. Porter, Diffusion-limited reaction rate theory for \ntwo-dimensional systems. Proceedings of the Royal Society of London. A. Mathematical \nand Physical Sciences 387, 147-170 (1983). \n18. A. B. Koudriavtsev, R. F. Jameson, W. Linert, SpringerLink, The Law of Mass Action, \nEngineering Online Library (Springer Berlin Heidelberg : Imprint: Springer, Berlin, \nHeidelberg, ed. 1st 2001., 2001). \n19. L. Iversen et al., Ras activation by SOS: Allosteric regulation by altered fluctuation \ndynamics. Science 345, 50-54 (2014). \n20. P . Bandaru, Y . Kondo, J. Kuriyan, The Interdependent Activation of Son-of-Sevenless \nand Ras. Cold Spring Harb Perspect Med 9 (2019). \n21. R. N. Jorissen et al., Epidermal growth factor receptor: mechanisms of activation and \nsignalling. Experimental Cell Research 284, 31-53 (2003). \n22. H. Ren, A. A. Lee, L. J. N. Lew, J. B. DeGrandchamp, J. T. Groves, Positive feedback in \nRas activation by full-length SOS arises from autoinhibition release mechanism. \nBiophysical Journal 123, 3295-3303 (2024). \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted December 5, 2024. ; https://doi.org/10.1101/2024.12.04.626844doi: bioRxiv preprint \n\n23. J. Masison, P . J. Michalski, L. M. Loew, A. D. Schuyler, mol2sphere: spherical \ndecomposition of multi-domain molecules for visualization and coarse grained spatial \nmodeling. Bioinformatics 34, 3948-3950 (2018). \n24. J. Abramson et al., Accurate structure prediction of biomolecular interactions with \nAlphaFold 3. Nature 630, 493-500 (2024). \n25. J. Jumper et al., Highly accurate protein structure prediction with AlphaFold. Nature 596, \n583-589 (2021). \n26. G. I. Bell, Models for the Specific Adhesion of Cells to Cells. Science 200, 618-627 \n(1978). \n27. S. Hoops et al., COPASI: a COmplex PAthway SImulator. Bioinformatics 22, 3067-3074 \n(2006). \n28. J. Schaff, C. C. Fink, B. Slepchenko, J. H. Carson, L. M. Loew, A general computational \nframework for modeling cellular structure and function. Biophys J 73, 1135-1146 (1997). \n29. A. E. Cowan, Moraru, II, J. C. Schaff, B. M. Slepchenko, L. M. Loew, Spatial modeling of \ncell signaling networks. Methods Cell Biol 110, 195-221 (2012). \n \n \n \n \nFigure Captions \n \nFigure 1. Workflow for finding membrane bimolecular binding rate constants (𝑘𝑘\n𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚)) in \nterms of surface densities. Step 1 is to run 100 SpringSaLaD trajectories with the coarse \ngrained model of the binding partners. Illustrated are a pair of simple monomers (top) consisting \nof 2nm diameter spheres (yellow) tethered to a 1nm membrane anchor sphere (gray) with a \n5nm link; at the bottom, the product dimer is depicted. Step 2 consists of fitting the average of \n100 outputs from stochastic SpringSaLaD simulations to a deterministic (ODE) non-spatial \nmodel of surface-bound dimerization. \n \nFigure 2. Examples of SpringSaLaD simulation data and their fit to surface-confined \nmass action dimerization kinetics. The initial surface density of monomers in each case is \n2500 molecule/µm\n2. The diffusion coefficient of the anchor and the volumetric binding rate \nconstant of the binding sites are indicated above each graph, corresponding respectively to \nrows 1, 6 and 5 of Table 1. \n \nFigure 3. Membrane binding of Ras to the catalytic site of Receptor-bound SOS. Top: \ndirect binding. Bottom: after pre-binding of Ras to the allosteric site. The molecular \nstructures are approximated by using the SpringSaLaD 3D editing utility based on atomic \nstructures derived from AlphaFold2. The top left structure is an EGFR cytoplasmic domain \nanchored to the membrane (red kinase domain, followed by yellow disordered tail capped by a \nphosphotyrosine in green); the latter is linked to a cyan SH2 domain in Grb2; one of its magenta \nSH3 domains is linked to an olive PRM on the end of the disordered region of SOS; the violet \nSOS binding site for the catalytic domain of Ras is indicated with an asterisk (*). The bottom left \nstructure is identical, except that the pink allosteric site on SOS is bound to Ras. The Ras \nstructures are shown in the center with the yellow binding sites indicated by an asterisk. The \ninput rate constants for the SpringSaLaD simulations are shown at the top, corresponding to the \nvolumetric on rate for Ras binding to the catalytic site of SOS. The EGFR anchor diffusion \ncoefficient is 0.01µm\n2/s. All other site diffusion coefficients are 1.0µm2/s. For each condition, 20 \nEGFR-Grb2-SOS molecules react with 200 Ras molecules on a 250nmX250nm membrane \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted December 5, 2024. ; https://doi.org/10.1101/2024.12.04.626844doi: bioRxiv preprint \n\nsurface to generate 100 SpringSaLaD trajectories. Their means were fit to a deterministic 2D \nrate law to derive  𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚), fixing koff at 4.0 s-1; results for the 2 conditions are shown on the right. \n \n \nList of Supporting Data and Information \nDerivations for limiting cases related to Table 1. Filename: Supporting Text - Derivations for \nlimiting cases.docx \nSpringSaLaD input files for the models in Tables 1 and 2 and Figure 3. Filenames: Table 1 and 2 \nSIMS.zip; EGFR_Grb_SOS binding to Ras.txt; EGFR_Grb_SOS_prebound at allo binding to \nRas.txt \nSpreadsheets related to Tables 1 and 2 and Figure 3 containing the SpringSaLaD simulation \nresults and the fits to determine 𝑘𝑘𝑜𝑜𝑜𝑜\n(𝑚𝑚𝑚𝑚𝑚𝑚). Filenames: Table 1 5nm_stiff_fits.xlsx; Table2 \n20nmT_fits.xlsx; Data for Figure 3.xlsx \nMovie of an example trajectory for the model of Table 1 Row 5 and also Fig. 2, 3rd panel. \nFilename: Movie kon=1 Danchor=membrane protein.mp4 \n \n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted December 5, 2024. ; https://doi.org/10.1101/2024.12.04.626844doi: bioRxiv preprint","source_license":"CC-BY-4.0","license_restricted":false}