Dimensional Dependence of Binding Kinetics.

In the context of protein-protein binding, the dissociation constant is used to describe the affinity between two proteins. For protein-protein interactions, most experimentally-measured dissociation constants are measured in solution and reported in units of volume concentration. However, many protein interactions take place on membranes. These interactions have dissociation constants with units of areal concentration, rather than volume concentration. Here, we present a novel, stochastic approach to understanding the dimensional dependence of binding kinetics. Using stochastic exit time calculations, in discrete and continuous space, we derive general reaction rates for protein-protein binding in one, two, and three dimensions and demonstrate that dimensionality greatly affects binding kinetics. Further, we present a formula to transform three-dimensional experimentally-measured dissociation constants to two-dimensional dissociation constants. This conversion can be used to mathematically model binding events that occur on membranes.

Initiation and entrainment of multicellular automaticity via diffusion limited extracellular domains.

Electrically excitable cells often spontaneously and synchronously depolarize in vitro and in vivo preparations. It remains unclear how cells entrain and autorhythmically activate above the intrinsic mean activation frequency of isolated cells with or without pacemaking mechanisms. Recent studies suggest that cyclic ion accumulation and depletion in diffusion-limited extracellular volumes modulate electrophysiology by ephaptic mechanisms (nongap junction or synaptic coupling). This report explores how potassium accumulation and depletion in a restricted extracellular domain induces spontaneous action potentials in two different computational models of excitable cells without gap junctional coupling: Hodgkin-Huxley and Luo-Rudy. Importantly, neither model will spontaneously activate on its own without external stimuli. Simulations demonstrate that cells sharing a diffusion-limited extracellular compartment can become autorhythmic and entrained despite intercellular electrical heterogeneity. Autorhythmic frequency is modulated by the cleft volume and potassium fluxes through the cleft. Additionally, inexcitable cells can suppress or induce autorhythmic activity in an excitable cell via a shared cleft. Diffusion-limited shared clefts can also entrain repolarization. Critically, this model predicts a mechanism by which diffusion-limited shared clefts can initiate, entrain, and modulate multicellular automaticity in the absence of gap junctions.

The PafBC-mediated response sensitizes a bistable DNA damage response in Mycobacteria.

Due to the genotoxically challenging environments in which they live in, Mycobacteria have a complex DNA damage repair system that is governed by two major DNA damage responses, namely, the LexA/RecA-dependent response and the newly characterized PafBC-mediated response (Müller et al., 2018). The LexA/RecA-dependent response is a well-known bistable response found in different types of bacteria, and the Mycobacteria-specific PafBC-mediated response interacts with and modifies the LexA/RecA-dependent response (Müller et al., 2018). The interaction between the LexA/RecA-dependent response and the PafBC-mediated response has not been characterized mathematically. Our analysis shows that the addition of the PafBC-mediated response sensitizes the overall DNA damage response, effectively lowering the DNA damage rate threshold for activation.

A mathematical model of flagellar gene regulation and construction in Salmonella enterica.

Millions of people worldwide develop foodborne illnesses caused by Salmonella enterica (S. enterica) every year. The pathogenesis of S. enterica depends on flagella, which are appendages that the bacteria use to move through the environment. Interestingly, populations of genetically identical bacteria exhibit heterogeneity in the number of flagella. To understand this heterogeneity and the regulation of flagella quantity, we propose a mathematical model that connects the flagellar gene regulatory network to flagellar construction. A regulatory network involving more than 60 genes controls flagellar assembly. The most important member of the network is the master operon, flhDC, which encodes the FlhD4C2 protein. FlhD4C2 controls the construction of flagella by initiating the production of hook basal bodies (HBBs), protein structures that anchor the flagella to the bacterium. By connecting a model of FlhD4C2 regulation to a model of HBB construction, we investigate the roles of various feedback mechanisms. Analysis of our model suggests that a combination of regulatory mechanisms at the protein and transcriptional levels induce bistable FlhD4C2 levels and heterogeneous numbers of flagella. Also, the balance of regulatory mechanisms that become active following HBB construction is sufficient to provide a counting mechanism for controlling the total number of flagella produced.

Bifurcation Analysis of a Heat-Sensitive Epigenetic Regulatory Network.

An epigenetic regulatory network that influences transgenerational inheritance of a heat-altered phenotype was recently discovered in Arabidopsis. Our analysis shows that transgenerational inheritance of the heat-altered phenotype operates in a switch-like manner and can be turned on or off as a function of heat. We also show that trans-acting small interfering RNAs act as an "inverse amplifier" of HTT5, the protein that controls the heat-altered phenotype by a currently unknown mechanism. Our analysis uses the resultant to find an analytic expression for a cusp curve in parameter space and to find a parameter bound on switch-like behavior.

A systematic search for switch-like behavior in type II toxin-antitoxin systems.

Bistable switch-like behavior is a ubiquitous feature of gene regulatory networks with decision-making capabilities. Type II toxin-antitoxin (TA) systems are hypothesized to facilitate a bistable switch in toxin concentration that influences the dormancy transition in persister cells. However, a series of recent retractions has raised fundamental questions concerning the exact mechanism of toxin propagation in persister cells and the relationship between type II TA systems and cellular dormancy. Through a careful modeling search, we identify how sp: bistablilty can emerge in type II TA systems by systematically modifying a basic model for the RelBE system with other common biological mechanisms. Our systematic search uncovers a new combination of mechanisms influencing bistability in type II TA systems and explores how toxin bistability emerges through synergistic interactions between paired type II TA systems. Our analysis also illustrates how Descartes' rule of signs and the resultant can be used as a powerful delineator of bistability in mathematical systems regardless of application.

A Mathematical Model for Inheritance of DNA Methylation Patterns in Somatic Cells.

DNA methylation is an essential epigenetic mechanism used by cells to regulate gene expression. Interestingly, DNA replication, a function necessary for cell division, disrupts the methylation pattern. Since perturbed methylation patterns are associated with aberrant gene expression and many diseases, including cancer, restoration of the correct pattern following DNA replication is crucial. However, the exact mechanisms of this restoration remain under investigation. DNA methyltransferases (Dnmts) perform methylation by adding a methyl group to cytosines at CpG sites in the DNA. These CpG sites are found in regions of high density, termed CpG islands (CGIs), and regions of low density in the genome. Nearly, every CpG site in a CGI has the same state, either methylated or unmethylated, and almost all CpG sites in regions of low CpG density are methylated. We propose a stochastic model for the dynamics of the post-replicative restoration of methylation patterns. The model considers the recruitment of Dnmts and demethylating enzymes to regions of hyper- and hypomethylation, respectively. The model also includes the interaction between Dnmt1 and PCNA, an enzyme that localizes Dnmt1 to the replication complex. Using our model, we predict that the methylation of regions of DNA can be bistable. Further, we predict that recruitment mechanisms maintain methylation in CGIs, whereas the Dnmt1-PCNA interaction maintains methylation in low-density regions.

Enhanced Nucleocytoplasmic Transport due to Competition for Elastic Binding Sites.

Nuclear pore complexes (NPCs) control all traffic into and out of the cell nucleus. NPCs are molecular machines that simultaneously achieve high selectivity and high transport rates. The biophysical details of how cargoes rapidly traverse the pore remain unclear but are known to be mediated by interactions between cargo-binding chaperone proteins and natively unstructured nucleoporin proteins containing many phenylalanine-glycine repeats (FG nups) that line the pore's central channel. Here, we propose a specific and detailed physical mechanism for the high speed of nuclear import based on the elasticity of FG nups and on competition between individual chaperone proteins for FG nup binding. We develop a mathematical model to support our proposed mechanism. We suggest that the recycling of nuclear import factors back to the cytoplasm is important for driving high-speed import and predict the existence of an optimal cytoplasmic concentration of cargo for enhancing the rate of import over a purely diffusive rate.

The computation of biomarkers in pharmacokinetics with the aid of singular perturbation methods.

In pharmacokinetics, exact solutions to one-compartment models with nonlinear elimination kinetics cannot be found analytically, if dosages are assumed to be administered repetitively through extravascular routes (Tang and Xiao in J Pharmacokinet Pharmacodyn 34(6):807-827, 2007). Hence, for the corresponding impulsed dynamical system, alternative methods need to be developed to find approximate solutions. The primary purpose of this paper is to use the method of matched asymptotic expansions (Holmes Introduction to Perturbation Methods, vol 20. Springer Science & Business Media, Berlin, 2012), a singular perturbation method (Holmes, Introduction to Perturbation Methods, vol 20. Springer Science & Business Media, Berlin, 2012; Keener Principles of Applied Mathematics, Addison-Wesley, Boston, 1988), to obtain approximate solutions. With this method, we are able to rigorously determine conditions under which there is a stable periodic solution of the model equations. Furthermore, typical important biomarkers that enable the design of practical, efficient and safe drug delivery protocols, such as the time the drug concentration reaches the peak and the peak concentrations, are theoretically estimated by the perturbation method we employ.

A physics-based model for maintenance of the pH gradient in the gastric mucus layer.

It is generally accepted that the gastric mucus layer provides a protective barrier between the lumen and the mucosa, shielding the mucosa from acid and digestive enzymes and preventing autodigestion of the stomach epithelium. However, the precise mechanisms that contribute to this protective function are still up for debate. In particular, it is not clear what physical processes are responsible for transporting hydrogen protons, secreted within the gastric pits, across the mucus layer to the lumen without acidifying the environment adjacent to the epithelium. One hypothesis is that hydrogen may be bound to the mucin polymers themselves as they are convected away from the mucosal surface and eventually degraded in the stomach lumen. It is also not clear what mechanisms prevent hydrogen from diffusing back toward the mucosal surface, thereby lowering the local pH. In this work we investigate a physics-based model of ion transport within the mucosal layer based on a Nernst-Planck-like equation. Analysis of this model shows that the mechanism of transporting protons bound to the mucus gel is capable of reproducing the trans-mucus pH gradients reported in the literature. Furthermore, when coupled with ion exchange at the epithelial surface, our analysis shows that bicarbonate secretion alone is capable of neutralizing the epithelial pH, even in the face of enormous diffusive gradients of hydrogen. Maintenance of the pH gradient is found to be robust to a wide array of perturbations in both physiological and phenomenological model parameters, suggesting a robust physiological control mechanism.NEW & NOTEWORTHY This work combines modeling techniques based on physical principles, as well as novel numerical simulations to test the plausibility of one hypothesized mechanism for proton transport across the gastric mucus layer. Results show that this mechanism is able to maintain the extreme pH gradient seen in in vivo experiments and suggests a highly robust regulation mechanism to maintain this gradient in the face of dynamic lumen composition.

Rebinding in biochemical reactions on membranes.

The behavior of many biochemical processes depends crucially on molecules rapidly rebinding after dissociating. In the case of multisite protein modification, the importance of rebinding has been demonstrated both experimentally and through several recent computational studies involving stochastic spatial simulations. As rebinding stems from spatio-temporal correlations, theorists have resorted to models that explicitly include space to properly account for the effects of rebinding. However, for reactions in three space dimensions it was recently shown that well-mixed ordinary differential equation (ODE) models can incorporate rebinding by adding connections to the reaction network. The rate constants for these new connections involve the probability that a pair of molecules rapidly rebinds after dissociation. In order to study biochemical reactions on membranes, in this paper we derive an explicit formula for this rebinding probability for reactions in two space dimensions. We show that ODE models can use the formula to replicate detailed stochastic spatial simulations, and that the formula can predict ultrasensitivity for reactions involving multisite modification of membrane-bound proteins. Further, we compute a new concentration-dependent rebinding probability for reactions in three space dimensions. Our analysis predicts that rebinding plays a much larger role in reactions on membranes compared to reactions in cytoplasm.

Bidirectionality from cargo thermal fluctuations in motor-mediated transport.

Molecular motor proteins serve as an essential component of intracellular transport by generating forces to haul cargoes along cytoskeletal filaments. Two species of motors that are directed oppositely (e.g. kinesin, dynein) can be attached to the same cargo, which is known to produce bidirectional net motion. Although previous work focuses on the motor number as the driving noise source for switching, we propose an alternative mechanism: cargo diffusion. A mean-field mathematical model of mechanical interactions of two populations of molecular motors with cargo thermal fluctuations (diffusion) is presented to study this phenomenon. The delayed response of a motor to fluctuations in the cargo velocity is quantified, allowing for the reduction of the full model a single "characteristic distance", a proxy for the net force on the cargo. The system is then found to be metastable, with switching exclusively due to cargo diffusion between distinct directional transport states. The time to switch between these states is then investigated using a mean first passage time analysis. The switching time is found to be non-monotonic in the drag of the cargo, providing an experimental test of the theory.

Including Rebinding Reactions in Well-Mixed Models of Distributive Biochemical Reactions.

The behavior of biochemical reactions requiring repeated enzymatic substrate modification depends critically on whether the enzymes act processively or distributively. Whereas processive enzymes bind only once to a substrate before carrying out a sequence of modifications, distributive enzymes release the substrate after each modification and thus require repeated bindings. Recent experimental and computational studies have revealed that distributive enzymes can act processively due to rapid rebindings (so-called quasi-processivity). In this study, we derive an analytical estimate of the probability of rapid rebinding and show that well-mixed ordinary differential equation models can use this probability to quantitatively replicate the behavior of spatial models. Importantly, rebinding requires that connections be added to the well-mixed reaction network; merely modifying rate constants is insufficient. We then use these well-mixed models to suggest experiments to 1) detect quasi-processivity and 2) test the theory. Finally, we show that rapid rebindings drastically alter the reaction's Michaelis-Menten rate equations.

Binding to any ESCRT can mediate ubiquitin-independent cargo sorting.

The ESCRT (endosomal sorting complex required for transport) machinery is known to sort ubiquitinated transmembrane proteins into vesicles that bud into the lumen of multivesicular bodies (MVBs). Although the ESCRTs themselves are ubiquitinated they are excluded from the intraluminal vesicles and recycle back to the cytoplasm for further rounds of sorting. To obtain insights into the rules that distinguish ESCRT machinery from cargo we analyzed the trafficking of artificial ESCRT-like protein fusions. These studies showed that lowering ESCRT-binding affinity converts a protein from behaving like ESCRT machinery into cargo of the MVB pathway, highlighting the close relationship between machinery and the cargoes they sort. Furthermore, our findings give insights into the targeting of soluble proteins into the MVB pathway and show that binding to any of the ESCRTs can mediate ubiquitin-independent MVB sorting.

Potassium channels in the Cx43 gap junction perinexus modulate ephaptic coupling: an experimental and modeling study.

It was recently demonstrated that cardiac sodium channels (Nav1.5) localized at the perinexus, an intercalated disc (ID) nanodomain associated with gap junctions (GJ), may contribute to electrical coupling between cardiac myocytes via an ephaptic mechanism. Impairment of ephaptic coupling by acute interstitial edema (AIE)-induced swelling of the perinexus was associated with arrhythmogenic, anisotropic conduction slowing. Given that Kir2.1 has also recently been reported to localize at intercalated discs, we hypothesized that Kir2.1 channels may reside within the perinexus and that inhibiting them may mitigate arrhythmogenic conduction slowing observed during AIE. Using gated stimulated emission depletion (gSTED) and stochastic optical reconstruction microscopy (STORM) super-resolution microscopy, we indeed find that a significant proportion of Kir2.1 channels resides within the perinexus. Moreover, whereas Nav1.5 inhibition during AIE exacerbated arrhythmogenic conduction slowing, inhibiting Kir2.1 channels during AIE preferentially increased transverse conduction velocity-decreasing anisotropy and ameliorating arrhythmia risk compared to AIE alone. Comparison of our results with a nanodomain computer model identified enrichment of both Nav1.5 and Kir2.1 at intercalated discs as key factors underlying the experimental observations. We demonstrate that Kir2.1 channels are localized within the perinexus alongside Nav1.5 channels. Further, targeting Kir2.1 modulates intercellular coupling between cardiac myocytes, anisotropy of conduction, and arrhythmia propensity in a manner consistent with a role for ephaptic coupling in cardiac conduction. For over half a century, electrical excitation in the heart has been thought to occur exclusively via gap junction-mediated ionic current flow between cells. Further, excitation was thought to depend almost exclusively on sodium channels with potassium channels being involved mainly in returning the cell to rest. Here, we demonstrate that sodium and potassium channels co-reside within nanoscale domains at cell-to-cell contact sites. Experimental and computer modeling results suggest a role for these channels in electrical coupling between cardiac muscle cells via an ephaptic mechanism working in tandem with gap junctions. This new insight into the mechanism of cardiac electrical excitation could pave the way for novel therapies against cardiac rhythm disturbances.

Sodium channels in the Cx43 gap junction perinexus may constitute a cardiac ephapse: an experimental and modeling study.

It has long been held that electrical excitation spreads from cell-to-cell in the heart via low resistance gap junctions (GJ). However, it has also been proposed that myocytes could interact by non-GJ-mediated "ephaptic" mechanisms, facilitating propagation of action potentials in tandem with direct GJ-mediated coupling. We sought evidence that such mechanisms contribute to cardiac conduction. Using super-resolution microscopy, we demonstrate that Nav1.5 is localized within 200 nm of the GJ plaque (a region termed the perinexus). Electron microscopy revealed close apposition of adjacent cell membranes within perinexi suggesting that perinexal sodium channels could function as an ephapse, enabling ephaptic cell-to-cell transfer of electrical excitation. Acute interstitial edema (AIE) increased intermembrane distance at the perinexus and was associated with preferential transverse conduction slowing and increased spontaneous arrhythmia incidence. Inhibiting sodium channels with 0.5 µM flecainide uniformly slowed conduction, but sodium channel inhibition during AIE slowed conduction anisotropically and increased arrhythmia incidence more than AIE alone. Sodium channel inhibition during GJ uncoupling with 25 µM carbenoxolone slowed conduction anisotropically and was also highly proarrhythmic. A computational model of discretized extracellular microdomains (including ephaptic coupling) revealed that conduction trends associated with altered perinexal width, sodium channel conductance, and GJ coupling can be predicted when sodium channel density in the intercalated disk is relatively high. We provide evidence that cardiac conduction depends on a mathematically predicted ephaptic mode of coupling as well as GJ coupling. These data suggest opportunities for novel anti-arrhythmic therapies targeting noncanonical conduction pathways in the heart.

Extracellular sodium and potassium levels modulate cardiac conduction in mice heterozygous null for the Connexin43 gene.

UNLABELLED: Several studies have disagreed on measurements of cardiac conduction velocity (CV) in mice with a heterozygous knockout of the connexin gene Gja1--a mutation that reduces the gap junction (GJ) protein, Connexin43 (Cx43), by 50 %. We noted that perfusate ionic composition varied between studies and hypothesized that extracellular ionic concentration modulates CV dependence on GJs. CV was measured by optically mapping wild-type (WT) and heterozygous null (HZ) hearts serially perfused with solutions previously associated with no change (Solution 1) or CV slowing (Solution 2). In WT hearts, CV was similar for Solutions 1 and 2. However, consistent with the hypothesis, Solution 2 in HZ hearts slowed transverse CV (CVT) relative to Solution 1. Previously, we showed CV slowing in a manner consistent with ephaptic conduction correlated with increased perinexal inter-membrane width (W P) at GJ edges. Thus, W P was measured following perfusion with systematically adjusted [Na(+)]o and [K(+)]o in Solutions 1 and 2. A wider W P was associated with reduced CVT in WT and HZ hearts, with the greatest effect in HZ hearts. Increasing [Na(+)]o increased CVT only in HZ hearts. Increasing [K(+)]o slowed CVT in both WT and HZ hearts with large W P but only in HZ hearts with narrow W P. CONCLUSION: When perinexi are wide, decreasing excitability by modulating [Na(+)]o and [K(+)]o increases CV sensitivity to reduced Cx43. By contrast, CV is less sensitive to Cx43 and ion composition when perinexi are narrow. These results are consistent with cardiac conduction dependence on both GJ and non-GJ (ephaptic) mechanisms.

An infrequent molecular ruler controls flagellar hook length in Salmonella enterica.

The bacterial flagellum consists of a long external filament connected to a membrane-embedded basal body at the cell surface by a short curved structure called the hook. In Salmonella enterica, the hook extends 55 nm from the cell surface. FliK, a secreted molecular ruler, controls hook length. Upon hook completion, FliK induces a secretion-specificity switch to filament-type substrate secretion. Here, we demonstrate that an infrequent ruler mechanism determines hook length. FliK is intermittently secreted during hook polymerization. The probability of the specificity switch is an increasing function of hook length. By uncoupling hook polymerization from FliK expression, we illustrate that FliK secretion immediately triggers the specificity switch in hooks greater than the physiological length. The experimental data display excellent agreement with a mathematical model of the infrequent ruler hypothesis. Merodiploid bacteria expressing simultaneously short and long ruler variants displayed hook-length control by the short ruler, further supporting the infrequent ruler model. Finally, the velocity of FliK secretion determines the probability of a productive FliK interaction with the secretion apparatus to change secretion substrate specificity.

Microdomain effects on transverse cardiac propagation.

The effect of gap junctional coupling, sodium ion channel distribution, and extracellular conductivity on transverse conduction in cardiac tissue is explored using a microdomain model that incorporates aspects of the inhomogeneous cellular structure. The propagation velocities found in our model are compared to those in the classic bidomain model and indicate a strong ephaptic microdomain contribution to conduction depending on the parameter regime. We show that ephaptic effects can be quite significant in the junctional spaces between cells, and that the cell activation sequence is modified substantially by these effects. Further, we find that transverse propagation can be maintained by ephaptic effects, even in the absence of gap junctional coupling. The mechanism by which this occurs is found to be cablelike in that the junctional regions act like inverted cables. Our results provide insight into several recent experimental studies that indirectly indicate a mode of action potential propagation that does not rely exclusively on gap junctions.

A mathematical model of force generation by flexible kinetochore-microtubule attachments.

Important mechanical events during mitosis are facilitated by the generation of force by chromosomal kinetochore sites that attach to dynamic microtubule tips. Several theoretical models have been proposed for how these sites generate force, and molecular diffusion of kinetochore components has been proposed as a key component that facilitates kinetochore function. However, these models do not explicitly take into account the recently observed flexibility of kinetochore components and variations in microtubule shape under load. In this paper, we develop a mathematical model for kinetochore-microtubule connections that directly incorporates these two important components, namely, flexible kinetochore binder elements, and the effects of tension load on the shape of shortening microtubule tips. We compare our results with existing biased diffusion models and explore the role of protein flexibility inforce generation at the kinetochore-microtubule junctions. Our model results suggest that kinetochore component flexibility and microtubule shape variation under load significantly diminish the need for high diffusivity (or weak specific binding) of kinetochore components; optimal kinetochore binder stiffness regimes are predicted by our model. Based on our model results, we suggest that the underlying principles of biased diffusion paradigm need to be reinterpreted.