Effect of surface curvature on diffusion-limited reactions on a curved surface

Slowest first passage times, redundancy, and menopause timing

Biological events are often initiated when a random "searcher" finds a "target," which is called a first passage time (FPT). In some biological systems involving multiple searchers, an important timescale is the time it takes the slowest searcher(s) to find a target. For example, of the hundreds of thousands of primordial follicles in a woman's ovarian reserve, it is the slowest to leave that trigger the onset of menopause. Such slowest FPTs may also contribute to the reliability of cell signaling pathways and influence the ability of a cell to locate an external stimulus. In this paper, we use extreme value theory and asymptotic analysis to obtain rigorous approximations to the full probability distribution and moments of slowest FPTs. Though the results are proven in the limit of many searchers, numerical simulations reveal that the approximations are accurate for any number of searchers in typical scenarios of interest. We apply these general mathematical results to models of ovarian aging and menopause timing, which reveals the role of slowest FPTs for understanding redundancy in biological systems. We also apply the theory to several popular models of stochastic search, including search by diffusive, subdiffusive, and mortal searchers.

Pore structure controls stability and molecular flux in engineered protein cages

Protein cages are a common architectural motif used by living organisms to compartmentalize and control biochemical reactions. While engineered protein cages have featured in the construction of nanoreactors and synthetic organelles, relatively little is known about the underlying molecular parameters that govern stability and flux through their pores. In this work, we systematically designed 24 variants of the Thermotoga maritima encapsulin cage, featuring pores of different sizes and charges. Twelve pore variants were successfully assembled and purified, including eight designs with exceptional thermal stability. While negatively charged mutations were better tolerated, we were able to form stable assemblies covering a full range of pore sizes and charges, as observed in seven new cryo-EM structures at 2.5- to 3.6-Å resolution. Molecular dynamics simulations and stopped-flow experiments revealed the importance of considering both pore size and charge, together with flexibility and rate-determining steps, when designing protein cages for controlling molecular flux.

Pore structure controls stability and molecular flux in engineered protein cages

Protein cages are a common architectural motif used by living organisms to compartmentalize and control biochemical reactions. While engineered protein cages have featured in the construction of nanoreactors and synthetic organelles, relatively little is known about the underlying molecular parameters that govern stability and flux through their pores. In this work, we systematically designed 24 variants of the Thermotoga maritima encapsulin cage, featuring pores of different sizes and charges. Twelve pore variants were successfully assembled and purified, including eight designs with exceptional thermal stability. While negatively charged mutations were better tolerated, we were able to form stable assemblies covering a full range of pore sizes and charges, as observed in seven new cryo-EM structures at 2.5- to 3.6-Å resolution. Molecular dynamics simulations and stopped-flow experiments revealed the importance of considering both pore size and charge, together with flexibility and rate-determining steps, when designing protein cages for controlling molecular flux.

Revising Berg-Purcell for finite receptor kinetics

From nutrient uptake to chemoreception to synaptic transmission, many systems in cell biology depend on molecules diffusing and binding to membrane receptors. Mathematical analysis of such systems often neglects the fact that receptors process molecules at finite kinetic rates. A key example is the celebrated formula of Berg and Purcell for the rate that cell surface receptors capture extracellular molecules. Indeed, this influential result is only valid if receptors transport molecules through the cell wall at a rate much faster than molecules arrive at receptors. From a mathematical perspective, ignoring receptor kinetics is convenient because it makes the diffusing molecules independent. In contrast, including receptor kinetics introduces correlations between the diffusing molecules because, for example, bound receptors may be temporarily blocked from binding additional molecules. In this work, we present a modeling framework for coupling bulk diffusion to surface receptors with finite kinetic rates. The framework uses boundary homogenization to couple the diffusion equation to nonlinear ordinary differential equations on the boundary. We use this framework to derive an explicit formula for the cellular uptake rate and show that the analysis of Berg and Purcell significantly overestimates uptake in some typical biophysical scenarios. We confirm our analysis by numerical simulations of a many-particle stochastic system.

Time matters for macroscopic membranes formed by alginate and cationic β-sheet peptides

Hierarchically ordered planar and spherical membranes (sacs) were constructed using amphiphilic and cationic β-sheet peptides that spontaneously assembled together with negatively charged alginate solution. The system was found to form either a fully developed membrane structure with three distinct regions including characteristic perpendicular fibers or a non-fully developed contact layer lacking these standing fibers, depending on the peptide age, membrane geometry and membrane incubation time. The morphological differences were found to strongly depend on fairly-long incubation time frames that influenced both the peptide's intrinsic alignment and the reaction-diffusion process taking place at the interface. A three-stage mechanism was suggested and key parameters affecting the development process were identified. Stability tests in biologically relevant buffers confirmed the suitability of these membranes for bio applications.

Diffusion-Limited Reaction Kinetics of a Reactant with Square Reactive Patches on a Plane

We present a simple reaction model to study the influence of the size, number, and spatial arrangement of reactive patches on a reactant placed on a plane. Specifically, we consider a reactant whose surface has an N × N square grid structure, with each square cell (or patch) being chemically reactive or inert for partner reactant molecules approaching the cell via diffusion. We calculate the rate constant for various cases with different reactive N × N square patterns using the finite element method. For N = 2, 3, we determine the reaction kinetics of all possible reactive patterns in the absence and presence of periodic boundary conditions, and from the analysis, we find that the dependences of the kinetics on the size, number, and spatial arrangement are similar to those observed in reactive patches on a sphere. Furthermore, using square reactant models, we present a method to significantly increase the rate constant by sequentially breaking the patches into smaller patches and arranging them symmetrically. Interestingly, we find that a reactant with a symmetric patch distribution has a power–law relation between the rate constant and the number of reactive patches and show that this works well when the total reactive area is much less than the total surface area of the reactant. Since our N × N discrete models enable us to examine all possible reactive cases completely, they provide a solid understanding of the surface reaction kinetics, which would be helpful for understanding the fundamental aspects of the competitions between reactive patches arising in real applications.

Effects of the Size, the Number, and the Spatial Arrangement of Reactive Patches on a Sphere on Diffusion-limited Reaction Kinetics: A Comprehensive Study

We investigate how the size, the number, and the spatial arrangement of identical nonoverlapping reactive patches on a sphere influence the overall reaction kinetics of bimolecular diffusion-limited (or diffusion-controlled) reactions that occur between the patches and the reactants diffusing around the sphere. First, in the arrangement of two patches, it is known that the overall rate constant increases as the two patches become more separated from each other but decreases when they become closer to each other. In this work, we further study the dependence of the patch arrangement on the kinetics with three and four patches using the finite element method (FEM). In addition to the patch arrangement, the kinetics is also dependent on the number and size of the patches. Therefore, we study such dependences by calculating the overall rate constants using the FEM for various cases, especially for large-sized patches, and this study is complementary to the kinetic studies that were performed by Brownian dynamics (BD) simulation methods for small-sized patches. The numerical FEM and BD simulation results are compared with the results from various kinetic theories to evaluate the accuracies of the theories. Remarkably, this comparison indicates that our theory, which was recently developed based on the curvature-dependent kinetic theory, shows good agreement with the FEM and BD numerical results. From this validation, we use our theory to further study the variation of the overall rate constant when the patches are arbitrarily arranged on a sphere. Our theory also confirms that to maximize the overall rate constant, we need to break large-sized patches into smaller-sized patches and arrange them to be maximally separated to reduce their competition.

Theory of Diffusion-Influenced Reaction Networks

A formalism is developed to describe how diffusion alters the kinetics of coupled reversible association-dissociation reactions in the presence of conformational changes that can modify the reactivity. The major difficulty in constructing a general theory is that, even to the lowest order, diffusion can change the structure of the rate equations of chemical kinetics by introducing new reaction channels (i.e., modifies the kinetic scheme). Therefore, the right formalism must be found that allows the influence of diffusion to be described in a concise and elegant way for networks of arbitrary complexity. Our key result is a set of non-Markovian rate equations involving stoichiometric matrices and net reaction rates (fluxes), in which these rates are coupled by a time-dependent pair association flux matrix, whose elements have a simple physical interpretation. Specifically, each element is the probability density that an isolated pair of reactants irreversibly associates at time t via one reaction channel on the condition that it started out with the dissociation products of another (or the same) channel. In the Markovian limit, the coupling of the chemical rates is described by committors (or splitting/capture probabilities). The committor is the probability that an isolated pair of reactants formed by dissociation at one site will irreversibly associate at another site rather than diffuse apart. We illustrate the use of our formalism by considering three reversible reaction schemes: (1) binding to a single site, (2) binding to two inequivalent sites, and (3) binding to a site whose reactivity fluctuates. In the first example, we recover the results published earlier, while in the second one we show that a new reaction channel appears, which directly connects the two bound states. The third example is particularly interesting because all species become coupled and an exchange-type bimolecular reaction appears. In the Markovian limit, some of the diffusion-modified rate constants that describe new transitions become negative, indicating that memory effects cannot be ignored.

Theory of curvature-dependent kinetics of diffusion-limited reactions and its application to ligand binding to a sphere with multiple receptors

We present a simple theory that explains how surface curvature affects the reaction kinetics of diffusion-limited reactions on spherically curved surfaces. In this theory, we derive a quadratic equation under the conditions that the rate constant satisfies the Hill and Smoluchowski rate constants at the lowest and highest curvatures, respectively, and that at a certain intermediate curvature, there should be a maximum value of the rate constant, which was recently found in our previous work. We find that the result obtained from our theory is in good agreement with the corresponding one obtained from numerical calculation. In addition, we show that our theory can be directly applied to the Šolc-Stockmayer model of axially symmetric reactants, which can be considered as a spherical reactant with a single reaction site. Furthermore, we discuss using our theory to improve the formula for the rate constant in the Berg-Purcell ligand-binding model of a cell membrane covered by multiple receptors. Our simple theory yields insight into the effect of curvature on diffusion-influenced reactions and provides a useful formula for easily and quantitatively evaluating the curvature effect.