Influence of neighboring reactive particles on diffusion-limited reactions

The generalized method of separation of variables for diffusion-influenced reactions: Irreducible Cartesian tensor technique

Motivated by the various applications of the trapping diffusion-influenced reaction theory in physics, chemistry, and biology, this paper deals with irreducible Cartesian tensor (ICT) technique within the scope of the generalized method of separation of variables (GMSV). We provide a survey from the basic concepts of the theory and highlight the distinctive features of our approach in contrast to similar techniques documented in the literature. The solution to the stationary diffusion equation under appropriate boundary conditions is represented as a series in terms of ICT. By means of proved translational addition theorem, we straightforwardly reduce the general boundary value diffusion problem for N spherical sinks to the corresponding resolving infinite set of linear algebraic equations with respect to the unknown tensor coefficients. These coefficients exhibit an explicit dependence on the arbitrary three-dimensional configurations of N sinks with different radii and surface reactivities. Our research contains all relevant mathematical details such as terminology, definitions, and geometrical structure, along with a step by step description of the GMSV algorithm with the ICT technique to solve the general diffusion boundary value problem within the scope of Smoluchowski's trapping model.

Crowding within synaptic junctions influences the degradation of nucleotides by CD39 and CD73 ectonucleotidases

Synapsed cells can communicate using exocytosed nucleotides like adenosine triphosphate (ATP). Ectonucleotidases localized to synaptic junctions degrade nucleotides into metabolites like adenosine monophosphate (AMP) or adenosine. Oftentimes nucleotide degradation occurs in a sequential manner, of which ATP degradation by CD39 and CD73 is a representative example. Here, CD39 first converts ATP and adenosine diphosphate (ADP) into AMP, after which AMP is dephosphorylated into adenosine by CD73. Hence, the concerted activity of CD39 and CD73 can help shape cellular responses to extracellular ATP. In a previous study, we demonstrated that coupled CD39 and CD73 activity within synapse-like junctions is strongly controlled by the enzymes' co-localization, their surface charge densities, and the electrostatic potential of the surrounding cell membranes. In this study, we demonstrate that crowders within synaptic junctions, which can include globular proteins like cytokines and membrane-bound proteins, impact coupled CD39 and CD73 ectonucleotidase activity and, in turn, the availability of intrasynapse ATP. Specifically, we developed a spatially explicit, reaction-diffusion model for the coupled conversion of ATP → AMP and AMP → adenosine in a model synaptic junction with crowders that is solved via the finite element method. Our modeling results suggest that the association rate for ATP to CD39 is strongly influenced by the density of intrasynaptic protein crowders, as increasing crowder density generally suppressed ATP association kinetics. Much of this suppression can be rationalized based on a loss of configurational entropy. The surface charges of crowders can further influence the association rate, with the surprising result that favorable crowder-nucleotide electrostatic interactions can yield CD39 association rates that are faster than crowder-free configurations. However, attractive crowder-nucleotide interactions decrease the rate and efficiency of adenosine production, which in turn increases the availability of ATP and AMP within the synapse relative to crowder-free configurations. These findings highlight how CD39 and CD73 ectonucleotidase activity, electrostatics, and crowding within synapses influence the availability of nucleotides for intercellular communication.

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.

Diffusion toward non-overlapping partially reactive spherical traps: Fresh insights onto classic problems

Several classic problems for particles diffusing outside an arbitrary configuration of non-overlapping partially reactive spherical traps in three dimensions are revisited. For this purpose, we describe the generalized method of separation of variables for solving boundary value problems of the associated modified Helmholtz equation. In particular, we derive a semi-analytical solution for the Green function that is the key ingredient to determine various diffusion-reaction characteristics such as the survival probability, the first-passage time distribution, and the reaction rate. We also present modifications of the method to determine numerically or asymptotically the eigenvalues and eigenfunctions of the Laplace operator and the Dirichlet-to-Neumann operator in such perforated domains. Some potential applications in chemical physics and biophysics are discussed, including diffusion-controlled reactions for mortal particles.

Co-localization and confinement of ecto-nucleotidases modulate extracellular adenosine nucleotide distributions

Nucleotides comprise small molecules that perform critical signaling roles in biological systems. Adenosine-based nucleotides, including adenosine tri-, di-, and mono-phosphate, are controlled through their rapid degradation by diphosphohydrolases and ecto-nucleotidases (NDAs). The interplay between nucleotide signaling and degradation is especially important in synapses formed between cells, which create signaling 'nanodomains'. Within these 'nanodomains', charged nucleotides interact with densely-packed membranes and biomolecules. While the contributions of electrostatic and steric interactions within such nanodomains are known to shape diffusion-limited reaction rates, less is understood about how these factors control the kinetics of nucleotidase activity. To quantify these factors, we utilized reaction-diffusion numerical simulations of 1) adenosine triphosphate (ATP) hydrolysis into adenosine monophosphate (AMP) and 2) AMP into adenosine (Ado) via two representative nucleotidases, CD39 and CD73. We evaluate these sequentially-coupled reactions in nanodomain geometries representative of extracellular synapses, within which we localize the nucleotidases. With this model, we find that 1) nucleotidase confinement reduces reaction rates relative to an open (bulk) system, 2) the rates of AMP and ADO formation are accelerated by restricting the diffusion of substrates away from the enzymes, and 3) nucleotidase co-localization and the presence of complementary (positive) charges to ATP enhance reaction rates, though the impact of these contributions on nucleotide pools depends on the degree to which the membrane competes for substrates. As a result, these contributions integratively control the relative concentrations and distributions of ATP and its metabolites within the junctional space. Altogether, our studies suggest that CD39 and CD73 nucleotidase activity within junctional spaces can exploit their confinement and favorable electrostatic interactions to finely control nucleotide signaling.

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.

Structural and dynamic basis of substrate permissiveness in hydroxycinnamoyltransferase (HCT)

Substrate permissiveness has long been regarded as the raw materials for the evolution of new enzymatic functions. In land plants, hydroxycinnamoyltransferase (HCT) is an essential enzyme of the phenylpropanoid metabolism. Although essential enzymes are normally associated with high substrate specificity, HCT can utilize a variety of non-native substrates. To examine the structural and dynamic basis of substrate permissiveness in this enzyme, we report the crystal structure of HCT from Selaginella moellendorffii and molecular dynamics (MD) simulations performed on five orthologous HCTs from several major lineages of land plants. Through altogether 17-μs MD simulations, we demonstrate the prevalent swing motion of an arginine handle on a submicrosecond timescale across all five HCTs, which plays a key role in native substrate recognition by these intrinsically promiscuous enzymes. Our simulations further reveal how a non-native substrate of HCT engages a binding site different from that of the native substrate and diffuses to reach the catalytic center and its co-substrate. By numerically solving the Smoluchowski equation, we show that the presence of such an alternative binding site, even when it is distant from the catalytic center, always increases the reaction rate of a given substrate. However, this increase is only significant for enzyme-substrate reactions heavily influenced by diffusion. In these cases, binding non-native substrates ‘off-center’ provides an effective rationale to develop substrate permissiveness while maintaining the native functions of promiscuous enzymes.

Examples abound of enzymes that can process substrates other than their native ones. However, the structural and dynamic basis of this promiscuity remains to be fully understood. In this work, we examine HCT, an intrinsically promiscuous acyltransferase with conserved function in all land plants. We uncover the sub-microsecond swing motion of a key arginine residue facilitating the recognition of both native and non-native substrates of HCT. We also quantify the impact of an off-center binding site on the non-native reaction rate. Although our calculations were inspired by HCT, the results apply in general, i.e., for enzymes heavily influenced by diffusion, binding non-native substrates ‘off-center’, even with rather weak affinity, can accelerate non-native reactions to appreciable levels.

Diffusion-influenced reaction rates for active "sphere-prolate spheroid" pairs and Janus dimers

The purpose of this paper is twofold. First, we provide a concise introduction to the generalized method of separation of variables for solving diffusion problems in canonical domains beyond conventional arrays of spheres. Second, as an important example of its application in the theory of diffusion-influenced reactions, we present an exact solution of the axially symmetric problem on diffusive competition in an array of two active particles (including Janus dumbbells) constructed of a prolate spheroid and a sphere. In particular, we investigate how the reaction rate depends on sizes of active particles, spheroid aspect ratio, particles' surface reactivity, and distance between their centers.

Product interactions and feedback in diffusion-controlled reactions

Steric or attractive interactions among reactants or between reactants and inert crowders can substantially influence the total rate of a diffusion-influenced reaction in the liquid phase. However, the role of the product species, which has typically different physical properties than the reactant species, has been disregarded so far. Here we study the effects of reactant-product and product-product interactions as well as asymmetric diffusion properties on the rate of diffusion-controlled reactions in the classical Smoluchowski-setup for chemical transformations at a perfect catalytic sphere. For this, we solve the diffusion equation with appropriate boundary conditions coupled by a mean-field approach on the second virial level to account for the particle interactions. We find that all particle spatial distributions and the total rate can change significantly, depending on the diffusion and interaction properties of the accumulated products. Complex competing and self-regulating (homeostatic) or self-amplifying effects are observed for the system, leading to both decrease and increase in the rates, as the presence of interacting products feeds back to the reactant flux and thus the rate with which the products are generated.

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

To investigate how the curvature of a reactive surface can affect reaction kinetics, we use a simple model in which a diffusion-limited bimolecular reaction occurs on a curved surface that is hollowed inward, flat, or extended outward while keeping the reactive area on the surface constant. By numerically solving the diffusion equation for this model using the finite element method, we find that the rate constant is a non-linear function of the surface curvature and that there is an optimal curvature providing the maximum value of the rate constant, which indicates that a spherical reactant whose entire surface is reactive (a uniformly reactive sphere) is not the most reactive species for a given reactive surface area. We discuss how this result arises from the interplay between two opposing effects: the exposedness of the reactive area to its partner reactants, which causes the rate constant to increase as the curvature increases, and the competition occurring on the reactive surface, which decreases the rate constant. This study helps us to understand the role of curvature in surface reactions and allows us to rationally design reactants that provide a high reaction rate.

Point-particle method to compute diffusion-limited cellular uptake

We present an efficient point-particle approach to simulate reaction-diffusion processes of spherical absorbing particles in the diffusion-limited regime, as simple models of cellular uptake. The exact solution for a single absorber is used to calibrate the method, linking the numerical parameters to the physical particle radius and uptake rate. We study the configurations of multiple absorbers of increasing complexity to examine the performance of the method by comparing our simulations with available exact analytical or numerical results. We demonstrate the potential of the method to resolve the complex diffusive interactions, here quantified by the Sherwood number, measuring the uptake rate in terms of that of isolated absorbers. We implement the method in a pseudospectral solver that can be generalized to include fluid motion and fluid-particle interactions. As a test case of the presence of a flow, we consider the uptake rate by a particle in a linear shear flow. Overall, our method represents a powerful and flexible computational tool that can be employed to investigate many complex situations in biology, chemistry, and related sciences.

An analysis of early-stage IL-2 capture times in populations of T cells diffusively interacting in a confined environment

This numerical analysis examines early-stage Interlukin-2 (IL-2) capture in large populations of secreting T helper (Th) and absorbing T regulatory (Treg) cells in an attempt to provide rational guidelines for when diffusive interactions can affect the Th autocrine cycle, as reflected in capture times. Autocrine and paracrine capture is calculated over a wide range of conditions: the mix of cells in a population; cell size and spacing; antigen activated IL-2 secretion and Th receptor expression rates; receptor dissociation constant; and number of resting Treg receptors. Correlations for quickly estimating IL-2 capture over these conditions are provided. This study suggests that a typical Treg can scavenge a significant amount of IL-2 without affecting autocrine capture by the Th. As a result, Treg influence on autocrine capture is shorter-ranged than previously reported. It is conjectured that high early-stage paracrine relative to autocrine capture leads to faster receptor enhancement for a Treg than a Th. The resulting enhancement time gap is considerably longer and, thus, diffusive suppression more likely, for a weakly- as opposed to strongly-activated Th. The methodology can be extended to later-stage capture to confirm this conjecture and to diffusive interactions in other cell-type populations.

Quantifying the Influence of the Crowded Cytoplasm on Small Molecule Diffusion

Cytosolic crowding can influence the thermodynamics and kinetics of in vivo chemical reactions. Most significantly, proteins and nucleic acid crowders reduce the accessible volume fraction, ϕ, available to a diffusing substrate, thereby reducing its effective diffusion rate, Deff, relative to its rate in bulk solution. However, Deff can be further hindered or even enhanced, when long-range crowder/diffuser interactions are significant. To probe these effects, we numerically estimated Deff values for small, charged molecules in representative, cytosolic protein lattices up to 0.1 × 0.1 × 0.1 μm(3) in volume via the homogenized Smoluchowski electro-diffusion equation. We further validated our predictions against Deff estimates from ϕ-dependent analytical relationships, such as the Maxwell-Garnett (MG) bound, as well as explicit solutions of the time-dependent electro-diffusion equation. We find that in typical, moderately crowded cell cytoplasm (ϕ ≈ 0.8), Deff is primarily determined by ϕ; in other words, diverse protein shapes and heterogeneous distributions only modestly impact Deff. However, electrostatic interactions between diffusers and crowders, particularly at low electrolyte ionic strengths, can substantially modulate Deff. These findings help delineate the extent that cytoplasmic crowders influence small molecule diffusion, which ultimately may shape the efficiency and timing of intracellular signaling pathways. More generally, the quantitative agreement between computationally expensive solutions of the time-dependent electro-diffusion equation and its comparatively cheaper homogenized form suggest that the latter is a broadly effective model for diffusion in wide-ranging, crowded biological media.

Theory of diffusion-influenced reactions in complex geometries

Chemical transformations involving the diffusion of reactants and subsequent chemical fixation steps are generally termed "diffusion-influenced reactions" (DIR). Virtually all biochemical processes in living media can be counted among them, together with those occurring in an ever-growing number of emerging nano-technologies. The role of the environment's geometry (obstacles, compartmentalization) and distributed reactivity (competitive reactants, traps) is key in modulating the rate constants of DIRs, and is therefore a prime design parameter. Yet, it is a formidable challenge to build a comprehensive theory that is able to describe the environment's "reactive geometry". Here we show that such a theory can be built by unfolding this many-body problem through addition theorems for special functions. Our method is powerful and general and allows one to study a given DIR reaction occurring in arbitrary "reactive landscapes", made of multiple spherical boundaries of given size and reactivity. Importantly, ready-to-use analytical formulas can be derived easily in most cases.

Enzyme localization, crowding, and buffers collectively modulate diffusion-influenced signal transduction: Insights from continuum diffusion modeling

Biochemical reaction networks consisting of coupled enzymes connect substrate signaling events with biological function. Substrates involved in these reactions can be strongly influenced by diffusion "barriers" arising from impenetrable cellular structures and macromolecules, as well as interactions with biomolecules, especially within crowded environments. For diffusion-influenced reactions, the spatial organization of diffusion barriers arising from intracellular structures, non-specific crowders, and specific-binders (buffers) strongly controls the temporal and spatial reaction kinetics. In this study, we use two prototypical biochemical reactions, a Goodwin oscillator, and a reaction with a periodic source/sink term to examine how a diffusion barrier that partitions substrates controls reaction behavior. Namely, we examine how conditions representative of a densely packed cytosol, including reduced accessible volume fraction, non-specific interactions, and buffers, impede diffusion over nanometer length-scales. We find that diffusion barriers can modulate the frequencies and amplitudes of coupled diffusion-influenced reaction networks, as well as give rise to "compartments" of decoupled reactant populations. These effects appear to be intensified in the presence of buffers localized to the diffusion barrier. These findings have strong implications for the role of the cellular environment in tuning the dynamics of signaling pathways.

Conformation-controlled binding kinetics of antibodies

Antibodies are large, extremely flexible molecules, whose internal dynamics is certainly key to their astounding ability to bind antigens of all sizes, from small hormones to giant viruses. In this paper, we build a shape-based coarse-grained model of IgG molecules and show that it can be used to generate 3D conformations in agreement with single-molecule Cryo-Electron Tomography data. Furthermore, we elaborate a theoretical model that can be solved exactly to compute the binding rate constant of a small antigen to an IgG in a prescribed 3D conformation. Our model shows that the antigen binding process is tightly related to the internal dynamics of the IgG. Our findings pave the way for further investigation of the subtle connection between the dynamics and the function of large, flexible multi-valent molecular machines.

Electrostatic channeling in P. falciparum DHFR-TS: Brownian dynamics and Smoluchowski modeling

We perform Brownian dynamics simulations and Smoluchowski continuum modeling of the bifunctional Plasmodium falciparum dihydrofolate reductase-thymidylate synthase (P. falciparum DHFR-TS) with the objective of understanding the electrostatic channeling of dihydrofolate generated at the TS active site to the DHFR active site. The results of Brownian dynamics simulations and Smoluchowski continuum modeling suggest that compared to Leishmania major DHFR-TS, P. falciparum DHFR-TS has a lower but significant electrostatic-mediated channeling efficiency (?15-25%) at physiological pH (7.0) and ionic strength (150 mM). We also find that removing the electric charges from key basic residues located between the DHFR and TS active sites significantly reduces the channeling efficiency of P. falciparum DHFR-TS. Although several protozoan DHFR-TS enzymes are known to have similar tertiary and quaternary structure, subtle differences in structure, active-site geometry, and charge distribution appear to influence both electrostatic-mediated and proximity-based substrate channeling.

Competitive effects between stationary chemical reaction centres: a theory based on off-center monopoles

The subject of this paper is competitive effects between multiple reaction sinks. A theory based on off-center monopoles is developed for the steady-state diffusion equation and for the convection-diffusion equation with a constant flow field. The dipolar approximation for the diffusion equation with two equal reaction centres is compared with the exact solution. The former turns out to be remarkably accurate, even for two touching spheres. Numerical evidence is presented to show that the same holds for larger clusters (with more than two spheres). The theory is extended to the convection-diffusion equation with a constant flow field. As one increases the convective velocity, the competitive effects between the reactive centres gradually become less significant. This is demonstrated for a number of cluster configurations. At high flow velocities, the current methodology breaks down. Fixing this problem will be the subject of future research. The current method is useful as an easy-to-use tool for the calibration of other more complicated models in mass and/or heat transfer.

A model study of sequential enzyme reactions and electrostatic channeling

We study models of two sequential enzyme-catalyzed reactions as a basic functional building block for coupled biochemical networks. We investigate the influence of enzyme distributions and long-range molecular interactions on reaction kinetics, which have been exploited in biological systems to maximize metabolic efficiency and signaling effects. Specifically, we examine how the maximal rate of product generation in a series of sequential reactions is dependent on the enzyme distribution and the electrostatic composition of its participant enzymes and substrates. We find that close proximity between enzymes does not guarantee optimal reaction rates, as the benefit of decreasing enzyme separation is countered by the volume excluded by adjacent enzymes. We further quantify the extent to which the electrostatic potential increases the efficiency of transferring substrate between enzymes, which supports the existence of electrostatic channeling in nature. Here, a major finding is that the role of attractive electrostatic interactions in confining intermediate substrates in the vicinity of the enzymes can contribute more to net reactive throughput than the directional properties of the electrostatic fields. These findings shed light on the interplay of long-range interactions and enzyme distributions in coupled enzyme-catalyzed reactions, and their influence on signaling in biological systems.

Molecular and subcellular-scale modeling of nucleotide diffusion in the cardiac myofilament lattice

Contractile function of cardiac cells is driven by the sliding displacement of myofilaments powered by the cycling myosin crossbridges. Critical to this process is the availability of ATP, which myosin hydrolyzes during the cross-bridge cycle. The diffusion of adenine nucleotides through the myofilament lattice has been shown to be anisotropic, with slower radial diffusion perpendicular to the filament axis relative to parallel, and is attributed to the periodic hexagonal arrangement of the thin (actin) and thick (myosin) filaments. We investigated whether atomistic-resolution details of myofilament proteins can refine coarse-grain estimates of diffusional anisotropy for adenine nucleotides in the cardiac myofibril, using homogenization theory and atomistic thin filament models from the Protein Data Bank. Our results demonstrate considerable anisotropy in ATP and ADP diffusion constants that is consistent with experimental measurements and dependent on lattice spacing and myofilament overlap. A reaction-diffusion model of the half-sarcomere further suggests that diffusional anisotropy may lead to modest adenine nucleotide gradients in the myoplasm under physiological conditions.