DEVELOPMENT OF FIBRIN BRANCH STRUCTURE BEFORE AND AFTER GELATION.

In [Fogelson and Keener, Phys. Rev. E, 81 (2010), 051922], we introduced a kinetic model of fibrin polymerization during blood clotting that captured salient experimental observations about how the gel branching structure depends on the conditions under which the polymerization occurs. Our analysis there used a moment-based approach that is valid only before the finite time blow-up that indicates formation of a gel. Here, we extend our analyses of the model to include both pre-gel and post-gel dynamics using the PDE-based framework we introduced in [Fogelson and Keener, SIAM J. Appl. Math., 75 (2015), pp. 1346-1368]. We also extend the model to include spatial heterogeneity and spatial transport processes. Studies of the behavior of the model reveal different spatial-temporal dynamics as the time scales of the key processes of branch formation, monomer introduction, and diffusion are varied.

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.

Modeling and Simulation of the Ion-Binding-Mediated Swelling Dynamics of Mucin-like Polyelectrolyte Gels.

Volume phase transitions in polyeletrolyte gels play important roles in many biophysical processes such as DNA packaging, nerve excitation, and cellular secretion. The swelling and deswelling of these charged polymer gels depend strongly on their ionic environment. In this paper, we present an extension to our previous two-fluid model for ion-binding-mediated gel swelling. The extended model eliminates the assumptions about the size similarity between the network and solvent particles, which makes it suitable for investigating of a large family of biologically relevant problems. The model treats the polyeletrolyte gel as a mixture of two materials, the network and the solvent. The dynamics of gel swelling is governed by the balance between the mechanical and chemical forces on each of these two materials. Simulations based on the model illustrate that the chemical forces are significantly influenced by the binding/unbinding reactions between the ions and the network, as well as the resulting distribution of charges within the gel. The dependence of the swelling rate on ionic bath concentrations is analyzed and this analysis highlights the importance of the electromigration of ions and the induced electric field in regulating gel swelling.

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.

Principles of slowed hydrogen diffusion through a mucus layer.

The control of transport through mucus layers is a ubiquitous phenomenon in physiological systems. Mucus is often tasked with the mediation of passive, diffusive transport of small ionic species. However, questions remain regarding how mucin gel characteristics (charge density of the polymeric network, binding affinity of ions with mucus) govern the rate at which ions diffuse through mucus layers. Experimental studies measuring hydrogen diffusion through gastric mucus have provided conflicting results, and it is not clear if the rate of ionic diffusion through mucus layers is appreciably different than in aqueous environments (depending on experimental preparation). Here, we present a mathematical analysis of electrodiffussion of two ionic species (hydrogen and chloride) through a mucus layer. In addition to accounting for the chemical binding of hydrogen to the mucus network, we enforce a zero net current condition (as mucus layers in physiological systems are not generally electrogenic) and calculate the Donnan potential that occurs at the edge of the mucus layer. The model predicts the steady-state fluxes of ionic species and the induced potential across the layer. We characterize the dependence of these quantities on the chemical properties of the mucus gel, the composition of the bath solution, and the molecular mobility of the dissolved anion, and we show that the model predictions are consistent with a large portion of the experimental literature. Our analysis predicts that mucus layers generically slow the diffusive transport of hydrogen, but that chemical binding with the network attenuates this effect.

Lumped parameter models for two-ventricle and healthy and failing extracardiac Fontan circulations.

Fontan circulations are surgical strategies to treat infants born with single ventricle physiology. Clinical and mathematical definitions of Fontan failure are lacking, and understanding is needed of parameters indicative of declining physiologies. Our objective is to develop lumped parameter models of two-ventricle and single-ventricle circulations. These models, their mathematical formulations and a proof of existence of periodic solutions are presented. Sensitivity analyses are performed to identify key parameters. Systemic venous and systolic left ventricular compliances and systemic capillary and pulmonary venous resistances are identified as key parameters. Our models serve as a framework to study the differences between two-ventricle and single-ventricle physiologies and healthy and failing Fontan circulations.

ENHANCED ELECTRODIFFUSIVE TRANSPORT ACROSS A MUCUS LAYER.

Diffusive transport of small ionic species through mucus layers is a ubiquitous phenomenon in physiology. However, some debate remains regarding how the various characteristics of mucus (charge of the polymers themselves, binding affinity of ions with mucus) impact the rate at which small ions may diffuse through a hydrated mucus gel. Indeed it is not even clear if small ionic species diffuse through mucus gel at an appreciably different rate than they do in aqueous solution. Here, we present a mathematical description of the transport of two ionic species (hydrogen and chloride) through a mucus layer based on the Nernst-Planck equations of electrodiffusion. The model explicitly accounts for the binding affinity of hydrogen to the mucus material, as well as the Donnan potential that occurs at the interface between regions with and without mucus. Steady state fluxes of ionic species are quantified, as are their dependencies on the chemical properties of the mucus gel and the composition of the bath solution. We outline a mechanism for generating enhanced diffusive flux of hydrogen across the gel region, and hypothesize how this mechanism may be relevant to the apparently contradictory experimental data in the literature.

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.

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.

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.

Kinetic model of two-monomer polymerization.

We propose a kinetic gelation model of polymer growth with two monomeric types that have distinct functionalities (reaction sites), and can polymerize using different reaction types. The heterotypic aggregation of two monomer types is modeled using a moment generating function approach by tracking the temporal evolution of a closed system of moment equations up until gelation. We investigate several scenarios of polymerization with two distinct monomers that differ in the types of reactions that can occur. We determine numerical and analytical conditions for finite time blow-up (the emergence of an oligomer of infinite size) that depend on initial conditions, reaction rates, and number of reaction sites per monomer.

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.

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.

Electrodiffusion-Mediated Swelling of a Two-Phase Gel Model of Gastric Mucus.

Gastric mucus gel is known to exhibit dramatic and unique swelling behaviors in response to the ionic composition of the hydrating solution. This swelling behavior is important in the maintenance of the mucus layer lining the stomach wall, as the layer is constantly digested by enzymes in the lumen, and must be replenished by new mucus that swells as it is secreted from the gastric wall. One hypothesis suggests that the condensed state of mucus at secretion is maintained by transient bonds with calcium that form crosslinks. These crosslinks are lost as monovalent cations from the environment displace divalent crosslinkers, leading to a dramatic change in the energy of the gel and inducing the swelling behavior. Previous modeling work has characterized the equilibrium behavior of polyelectrolyte gels that respond to calcium crosslinking. Here, we present an investigation of the dynamic swelling behavior of a polyelectrolytic gel model of mucus. In particular, we quantified the rate at which a globule of initially crosslinked gel swells when exposed to an ionic bath. The dependence of this swelling rate on several parameters was characterized. We observed that swelling rate has a non-monotone dependence on the molarity of the bath solution, with moderate concentrations of available sodium inducing the fastest swelling.

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.

Molecular and Physical Mechanisms of Fibrinolysis and Thrombolysis from Mathematical Modeling and Experiments.

Despite the common use of thrombolytic drugs, especially in stroke treatment, there are many conflicting studies on factors affecting fibrinolysis. Because of the complexity of the fibrinolytic system, mathematical models closely tied with experiments can be used to understand relationships within the system. When tPA is introduced at the clot or thrombus edge, lysis proceeds as a front. We developed a multiscale model of fibrinolysis that includes the main chemical reactions: the microscale model represents a single fiber cross-section; the macroscale model represents a three-dimensional fibrin clot. The model successfully simulates the spatial and temporal locations of all components and elucidates how lysis rates are determined by the interplay between the number of tPA molecules in the system and clot structure. We used the model to identify kinetic conditions necessary for fibrinolysis to proceed as a front. We found that plasmin regulates the local concentration of tPA through forced unbinding via degradation of fibrin and tPA release. The mechanism of action of tPA is affected by the number of molecules present with respect to fibrin fibers. The physical mechanism of plasmin action (crawling) and avoidance of inhibition is defined. Many of these new findings have significant implications for thrombolytic treatment.

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.

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.

Bacterial flagella grow through an injection-diffusion mechanism.

The bacterial flagellum is a self-assembling nanomachine. The external flagellar filament, several times longer than a bacterial cell body, is made of a few tens of thousands subunits of a single protein: flagellin. A fundamental problem concerns the molecular mechanism of how the flagellum grows outside the cell, where no discernible energy source is available. Here, we monitored the dynamic assembly of individual flagella using in situ labelling and real-time immunostaining of elongating flagellar filaments. We report that the rate of flagellum growth, initially ∼1,700 amino acids per second, decreases with length and that the previously proposed chain mechanism does not contribute to the filament elongation dynamics. Inhibition of the proton motive force-dependent export apparatus revealed a major contribution of substrate injection in driving filament elongation. The combination of experimental and mathematical evidence demonstrates that a simple, injection-diffusion mechanism controls bacterial flagella growth outside the cell.