The discrete-time Kermack-McKendrick model: A versatile and computationally attractive framework for modeling epidemics.

The COVID-19 pandemic has led to numerous mathematical models for the spread of infection, the majority of which are large compartmental models that implicitly constrain the generation-time distribution. On the other hand, the continuous-time Kermack-McKendrick epidemic model of 1927 (KM27) allows an arbitrary generation-time distribution, but it suffers from the drawback that its numerical implementation is rather cumbersome. Here, we introduce a discrete-time version of KM27 that is as general and flexible, and yet is very easy to implement computationally. Thus, it promises to become a very powerful tool for exploring control scenarios for specific infectious diseases such as COVID-19. To demonstrate this potential, we investigate numerically how the incidence-peak size depends on model ingredients. We find that, with the same reproduction number and the same initial growth rate, compartmental models systematically predict lower peak sizes than models in which the latent and the infectious period have fixed duration.

The effects of internal forces and membrane heterogeneity on three-dimensional cell shapes.

The shape of cells and the control thereof plays a central role in a variety of cellular processes, including endo- and exocytosis, cell division and cell movement. Intra- and extracellular forces control the shapes, and while the shape changes in some processes such as exocytosis are intracellularly-controlled and localized in the cell, movement requires force transmission to the environment, and the feedback from it can affect the cell shape and mode of movement used. The shape of a cell is determined by its cytoskeleton (CSK), and thus shape changes involved in various processes involve controlled remodeling of the CSK. While much is known about individual components involved in these processes, an integrated understanding of how intra- and extracellular signals are coupled to the control of the mechanical changes involved is not at hand for any of them. As a first step toward understanding the interaction between intracellular forces imposed on the membrane and cell shape, we investigate the role of distributed surrogates for cortical forces in producing the observed three-dimensional shapes. We show how different balances of applied forces lead to such shapes, that there are different routes to the same end state, and that state transitions between axisymmetric shapes need not all be axisymmetric. Examples of the force distributions that lead to protrusions are given, and the shape changes induced by adhesion of a cell to a surface are studied. The results provide a reference framework for developing detailed models of intracellular force distributions observed experimentally, and provide a basis for studying how movement of a cell in a tissue or fluid is influenced by its shape.

Visualizing mesoderm and neural crest cell dynamics during chick head morphogenesis.

Vertebrate head morphogenesis involves carefully-orchestrated tissue growth and cell movements of the mesoderm and neural crest to form the distinct craniofacial pattern. To better understand structural birth defects, it is important that we characterize the dynamics of these processes and learn how they rely on each other. Here we examine this question during chick head morphogenesis using time-lapse imaging, computational modeling, and experiments. We find that head mesodermal cells in culture move in random directions as individuals and move faster in the presence of neural crest cells. In vivo, mesodermal cells migrate in a directed manner and maintain neighbor relationships; neural crest cells travel through the mesoderm at a faster speed. The mesoderm grows with a non-uniform spatio-temporal profile determined by BrdU labeling during the period of faster and more-directed neural crest collective migration through this domain. We use computer simulations to probe the robustness of neural crest stream formation by varying the spatio-temporal growth profile of the mesoderm. We follow this with experimental manipulations that either stop mesoderm growth or prevent neural crest migration and observe changes in the non-manipulated cell population, implying a dynamic feedback between tissue growth and neural crest cell signaling to confer robustness to the system. Overall, we present a novel descriptive analysis of mesoderm and neural crest cell dynamics that reveals the coordination and co-dependence of these two cell populations during head morphogenesis.

A phosphoinositide-based model of actin waves in frustrated phagocytosis.

Phagocytosis is a complex process by which phagocytes such as lymphocytes or macrophages engulf and destroy foreign bodies called pathogens in a tissue. The process is triggered by the detection of antibodies that trigger signaling mechanisms that control the changes of the cellular cytoskeleton needed for engulfment of the pathogen. A mathematical model of the entire process would be extremely complicated, because the signaling and cytoskeletal changes produce large mechanical deformations of the cell. Recent experiments have used a confinement technique that leads to a process called frustrated phagocytosis, in which the membrane does not deform, but rather, signaling triggers actin waves that propagate along the boundary of the cell. This eliminates the large-scale deformations and facilitates modeling of the wave dynamics. Herein we develop a model of the actin dynamics observed in frustrated phagocytosis and show that it can replicate the experimental observations. We identify the key components that control the actin waves and make a number of experimentally-testable predictions. In particular, we predict that diffusion coefficients of membrane-bound species must be larger behind the wavefront to replicate the internal structure of the waves. Our model is a first step toward a more complete model of phagocytosis, and provides insights into circular dorsal ruffles as well.

A Random Walk Approach to Transport in Tissues and Complex Media: From Microscale Descriptions to Macroscale Models.

The biological processes necessary for the development and continued survival of any organism are often strongly influenced by the transport properties of various biologically active species. The transport phenomena involved vary over multiple temporal and spatial scales, from organism-level behaviors such as the search for food, to systemic processes such as the transport of oxygen from the lungs to distant organs, down to microscopic phenomena such as the stochastic movement of proteins in a cell. Each of these processes is influenced by many interrelated factors. Identifying which factors are the most important, and how they interact to determine the overall result is a problem of great importance and interest. Experimental observations are often fit to relatively simple models, but in reality the observations are the output of complicated functions of the physicochemical, topological, and geometrical properties of a given system. Herein we use multistate continuous-time random walks and generalized master equations to model transport processes involving spatial jumps, immobilization at defined sites, and stochastic internal state changes. The underlying spatial models, which are framed as graphs, may have different classes of nodes, and walkers may have internal states that are governed by a Markov process. A general form of the solutions, using Fourier-Laplace transforms and asymptotic analysis, is developed for several spatially infinite regular lattices in one and two spatial dimensions, and the theory is developed for the analysis of transport and internal state changes on general graphs. The goal in each case is to shed light on how experimentally observable macroscale transport coefficients can be explained in terms of microscale properties of the underlying processes. This work is motivated by problems arising in transport in biological tissues, but the results are applicable to a broad class of problems that arise in other applications.

A Model for the Hippo Pathway in the Drosophila Wing Disc.

Although significant progress has been made toward understanding morphogen-mediated patterning in development, control of the size and shape of tissues via local and global signaling is poorly understood. In particular, little is known about how cell-cell interactions are involved in the control of tissue size. The Hippo pathway in the Drosophila wing disc involves cell-cell interactions via cadherins, which lead to modulation of Yorkie, a cotranscriptional factor that affects control of the cell cycle and growth, and studies involving over- and underexpression of components of this pathway reveal conditions that lead to tissue over- or undergrowth. Here, we develop a mathematical model of the Hippo pathway that can qualitatively explain these observations, made in both whole-disc mutants and mutant-clone experiments. We find that a number of nonintuitive experimental results can be explained by subtle changes in the balances between inputs to the Hippo pathway and suggest some predictions that can be tested experimentally. We also show that certain components of the pathway are polarized at the single-cell level, which replicates observations of planar cell polarity. Because the signal transduction and growth control pathways are highly conserved between Drosophila and mammalian systems, the model we formulate can be used as a framework to guide future experimental work on the Hippo pathway in both Drosophila and mammalian systems.

Analysis of a model microswimmer with applications to blebbing cells and mini-robots.

Recent research has shown that motile cells can adapt their mode of propulsion depending on the environment in which they find themselves. One mode is swimming by blebbing or other shape changes, and in this paper we analyze a class of models for movement of cells by blebbing and of nano-robots in a viscous fluid at low Reynolds number. At the level of individuals, the shape changes comprise volume exchanges between connected spheres that can control their separation, which are simple enough that significant analytical results can be obtained. Our goal is to understand how the efficiency of movement depends on the amplitude and period of the volume exchanges when the spheres approach closely during a cycle. Previous analyses were predicated on wide separation, and we show that the speed increases significantly as the separation decreases due to the strong hydrodynamic interactions between spheres in close proximity. The scallop theorem asserts that at least two degrees of freedom are needed to produce net motion in a cyclic sequence of shape changes, and we show that these degrees can reside in different swimmers whose collective motion is studied. We also show that different combinations of mode sharing can lead to significant differences in the translation and performance of pairs of swimmers.

Getting in shape and swimming: the role of cortical forces and membrane heterogeneity in eukaryotic cells.

Recent research has shown that motile cells can adapt their mode of propulsion to the mechanical properties of the environment in which they find themselves-crawling in some environments while swimming in others. The latter can involve movement by blebbing or other cyclic shape changes, and both highly-simplified and more realistic models of these modes have been studied previously. Herein we study swimming that is driven by membrane tension gradients that arise from flows in the actin cortex underlying the membrane, and does not involve imposed cyclic shape changes. Such gradients can lead to a number of different characteristic cell shapes, and our first objective is to understand how different distributions of membrane tension influence the shape of cells in an inviscid quiescent fluid. We then analyze the effects of spatial variation in other membrane properties, and how they interact with tension gradients to determine the shape. We also study the effect of fluid-cell interactions and show how tension leads to cell movement, how the balance between tension gradients and a variable bending modulus determine the shape and direction of movement, and how the efficiency of movement depends on the properties of the fluid and the distribution of tension and bending modulus in the membrane.

Improving Parameter Inference from FRAP Data: an Analysis Motivated by Pattern Formation in the Drosophila Wing Disc.

Fluorescence recovery after photobleaching (FRAP) is used to obtain quantitative information about molecular diffusion and binding kinetics at both cell and tissue levels of organization. FRAP models have been proposed to estimate the diffusion coefficients and binding kinetic parameters of species for a variety of biological systems and experimental settings. However, it is not clear what the connection among the diverse parameter estimates from different models of the same system is, whether the assumptions made in the model are appropriate, and what the qualities of the estimates are. Here we propose a new approach to investigate the discrepancies between parameters estimated from different models. We use a theoretical model to simulate the dynamics of a FRAP experiment and generate the data that are used in various recovery models to estimate the corresponding parameters. By postulating a recovery model identical to the theoretical model, we first establish that the appropriate choice of observation time can significantly improve the quality of estimates, especially when the diffusion and binding kinetics are not well balanced, in a sense made precise later. Secondly, we find that changing the balance between diffusion and binding kinetics by changing the size of the bleaching region, which gives rise to different FRAP curves, provides a priori knowledge of diffusion and binding kinetics, which is important for model formulation. We also show that the use of the spatial information in FRAP provides better parameter estimation. By varying the recovery model from a fixed theoretical model, we show that a simplified recovery model can adequately describe the FRAP process in some circumstances and establish the relationship between parameters in the theoretical model and those in the recovery model. We then analyze an example in which the data are generated with a model of intermediate complexity and the parameters are estimated using models of greater or less complexity, and show how sensitivity analysis can be used to improve FRAP model formulation. Lastly, we show how sophisticated global sensitivity analysis can be used to detect over-fitting when using a model that is too complex.

Mechanisms of scaling in pattern formation.

Many organisms and their constituent tissues and organs vary substantially in size but differ little in morphology; they appear to be scaled versions of a common template or pattern. Such scaling involves adjusting the intrinsic scale of spatial patterns of gene expression that are set up during development to the size of the system. Identifying the mechanisms that regulate scaling of patterns at the tissue, organ and organism level during development is a longstanding challenge in biology, but recent molecular-level data and mathematical modeling have shed light on scaling mechanisms in several systems, including Drosophila and Xenopus. Here, we investigate the underlying principles needed for understanding the mechanisms that can produce scale invariance in spatial pattern formation and discuss examples of systems that scale during development.

Computational analysis of amoeboid swimming at low Reynolds number.

Recent experimental work has shown that eukaryotic cells can swim in a fluid as well as crawl on a substrate. We investigate the swimming behavior of Dictyostelium discoideum  amoebae who swim by initiating traveling protrusions at the front that propagate rearward. In our model we prescribe the velocity at the surface of the swimming cell, and use techniques of complex analysis to develop 2D models that enable us to study the fluid-cell interaction. Shapes that approximate the protrusions used by Dictyostelium discoideum  can be generated via the Schwarz-Christoffel transformation, and the boundary-value problem that results for swimmers in the Stokes flow regime is then reduced to an integral equation on the boundary of the unit disk. We analyze the swimming characteristics of several varieties of swimming Dictyostelium discoideum  amoebae, and discuss how the slenderness of the cell body and the shapes of the protrusion effect the swimming of these cells. The results may provide guidance in designing low Reynolds number swimming models.

A multi-time-scale analysis of chemical reaction networks: II. Stochastic systems.

We consider stochastic descriptions of chemical reaction networks in which there are both fast and slow reactions, and for which the time scales are widely separated. We develop a computational algorithm that produces the generator of the full chemical master equation for arbitrary systems, and show how to obtain a reduced equation that governs the evolution on the slow time scale. This is done by applying a state space decomposition to the full equation that leads to the reduced dynamics in terms of certain projections and the invariant distributions of the fast system. The rates or propensities of the reduced system are shown to be the rates of the slow reactions conditioned on the expectations of fast steps. We also show that the generator of the reduced system is a Markov generator, and we present an efficient stochastic simulation algorithm for the slow time scale dynamics. We illustrate the numerical accuracy of the approximation by simulating several examples. Graph-theoretic techniques are used throughout to describe the structure of the reaction network and the state-space transitions accessible under the dynamics.

A mathematical model of salt-sensitive hypertension: the neurogenic hypothesis.

Salt sensitivity of arterial pressure (salt-sensitive hypertension) is a serious global health issue. The causes of salt-sensitive hypertension are extremely complex and mathematical models can elucidate potential mechanisms that are experimentally inaccessible. Until recently, the only mathematical model for long-term control of arterial pressure was the model of Guyton and Coleman; referred to as the G-C model. The core of this model is the assumption that sodium excretion is driven by renal perfusion pressure, the so-called 'renal function curve'. Thus, the G-C model dictates that all forms of hypertension are due to a primary shift of the renal function curve to a higher operating pressure. However, several recent experimental studies in a model of hypertension produced by the combination of a high salt intake and administration of angiotensin II, the AngII-salt model, are inconsistent with the G-C model. We developed a new mathematical model that does not limit the cause of salt-sensitive hypertension solely to primary renal dysfunction. The model is the first known mathematical counterexample to the assumption that all salt-sensitive forms of hypertension require a primary shift of renal function: we show that in at least one salt-sensitive form of hypertension the requirement is not necessary. We will refer to this computational model as the 'neurogenic model'. In this Symposium Review we discuss how, despite fundamental differences between the G-C model and the neurogenic model regarding mechanisms regulating sodium excretion and vascular resistance, they generate similar haemodynamic profiles of AngII-salt hypertension. In addition, the steady-state relationships between arterial pressure and sodium excretion, a correlation that is often erroneously presented as the 'renal function curve', are also similar in both models. Our findings suggest that salt-sensitive hypertension is not due solely to renal dysfunction, as predicted by the G-C model, but may also result from neurogenic dysfunction.

The role of mathematical models in understanding pattern formation in developmental biology.

In a Wall Street Journal article published on April 5, 2013, E. O. Wilson attempted to make the case that biologists do not really need to learn any mathematics-whenever they run into difficulty with numerical issues, they can find a technician (aka mathematician) to help them out of their difficulty. He formalizes this in Wilsons Principle No. 1: "It is far easier for scientists to acquire needed collaboration from mathematicians and statisticians than it is for mathematicians and statisticians to find scientists able to make use of their equations." This reflects a complete misunderstanding of the role of mathematics in all sciences throughout history. To Wilson, mathematics is mere number crunching, but as Galileo said long ago, "The laws of Nature are written in the language of mathematics[Formula: see text] the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word." Mathematics has moved beyond the geometry-based model of Galileo's time, and in a rebuttal to Wilson, E. Frenkel has pointed out the role of mathematics in synthesizing the general principles in science (Both point and counter-point are available in Wilson and Frenkel in Notices Am Math Soc 60(7):837-838, 2013). We will take this a step further and show how mathematics has been used to make new and experimentally verified discoveries in developmental biology and how mathematics is essential for understanding a problem that has puzzled experimentalists for decades-that of how organisms can scale in size. Mathematical analysis alone cannot "solve" these problems since the validation lies at the molecular level, but conversely, a growing number of questions in biology cannot be solved without mathematical analysis and modeling. Herein, we discuss a few examples of the productive intercourse between mathematics and biology.

Constant-complexity stochastic simulation algorithm with optimal binning.

At the molecular level, biochemical processes are governed by random interactions between reactant molecules, and the dynamics of such systems are inherently stochastic. When the copy numbers of reactants are large, a deterministic description is adequate, but when they are small, such systems are often modeled as continuous-time Markov jump processes that can be described by the chemical master equation. Gillespie's Stochastic Simulation Algorithm (SSA) generates exact trajectories of these systems, but the amount of computational work required for each step of the original SSA is proportional to the number of reaction channels, leading to computational complexity that scales linearly with the problem size. The original SSA is therefore inefficient for large problems, which has prompted the development of several alternative formulations with improved scaling properties. We describe an exact SSA that uses a table data structure with event time binning to achieve constant computational complexity with respect to the number of reaction channels for weakly coupled reaction networks. We present a novel adaptive binning strategy and discuss optimal algorithm parameters. We compare the computational efficiency of the algorithm to existing methods and demonstrate excellent scaling for large problems. This method is well suited for generating exact trajectories of large weakly coupled models, including those that can be described by the reaction-diffusion master equation that arises from spatially discretized reaction-diffusion processes.

Stochastic analysis of reaction-diffusion processes.

Reaction and diffusion processes are used to model chemical and biological processes over a wide range of spatial and temporal scales. Several routes to the diffusion process at various levels of description in time and space are discussed and the master equation for spatially discretized systems involving reaction and diffusion is developed. We discuss an estimator for the appropriate compartment size for simulating reaction-diffusion systems and introduce a measure of fluctuations in a discretized system. We then describe a new computational algorithm for implementing a modified Gillespie method for compartmental systems in which reactions are aggregated into equivalence classes and computational cells are searched via an optimized tree structure. Finally, we discuss several examples that illustrate the issues that have to be addressed in general systems.

Emergent antibiotic persistence in a spatially structured synthetic microbial mutualism.

Antibiotic persistence (heterotolerance) allows a subpopulation of bacteria to survive antibiotic-induced killing and contributes to the evolution of antibiotic resistance. Although bacteria typically live in microbial communities with complex ecological interactions, little is known about how microbial ecology affects antibiotic persistence. Here, we demonstrated within a synthetic two-species microbial mutualism of Escherichia coli and Salmonella enterica that the combination of cross-feeding and community spatial structure can emergently cause high antibiotic persistence in bacteria by increasing the cell-to-cell heterogeneity. Tracking ampicillin-induced death for bacteria on agar surfaces, we found that E. coli forms up to 55 times more antibiotic persisters in the cross-feeding coculture than in monoculture. This high persistence could not be explained solely by the presence of S. enterica, the presence of cross-feeding, average nutrient starvation, or spontaneous resistant mutations. Time-series fluorescent microscopy revealed increased cell-to-cell variation in E. coli lag time in the mutualistic co-culture. Furthermore, we discovered that an E. coli cell can survive antibiotic killing if the nearby S. enterica cells on which it relies die first. In conclusion, we showed that the high antibiotic persistence phenotype can be an emergent phenomenon caused by a combination of cross-feeding and spatial structure. Our work highlights the importance of considering spatially structured interactions during antibiotic treatment and understanding microbial community resilience more broadly.

A hybrid model of tumor-stromal interactions in breast cancer.

Ductal carcinoma in situ (DCIS) is an early stage noninvasive breast cancer that originates in the epithelial lining of the milk ducts, but it can evolve into comedo DCIS and ultimately, into the most common type of breast cancer, invasive ductal carcinoma. Understanding the progression and how to effectively intervene in it presents a major scientific challenge. The extracellular matrix (ECM) surrounding a duct contains several types of cells and several types of growth factors that are known to individually affect tumor growth, but at present the complex biochemical and mechanical interactions of these stromal cells and growth factors with tumor cells is poorly understood. Here we develop a mathematical model that incorporates the cross-talk between stromal and tumor cells, which can predict how perturbations of the local biochemical and mechanical state influence tumor evolution. We focus on the EGF and TGF-ß signaling pathways and show how up- or down-regulation of components in these pathways affects cell growth and proliferation. We then study a hybrid model for the interaction of cells with the tumor microenvironment (TME), in which epithelial cells (ECs) are modeled individually while the ECM is treated as a continuum, and show how these interactions affect the early development of tumors. Finally, we incorporate breakdown of the epithelium into the model and predict the early stages of tumor invasion into the stroma. Our results shed light on the interactions between growth factors, mechanical properties of the ECM, and feedback signaling loops between stromal and tumor cells, and suggest how epigenetic changes in transformed cells affect tumor progression.

Excitation and adaptation in bacteria-a model signal transduction system that controls taxis and spatial pattern formation.

The machinery for transduction of chemotactic stimuli in the bacterium E. coli is one of the most completely characterized signal transduction systems, and because of its relative simplicity, quantitative analysis of this system is possible. Here we discuss models which reproduce many of the important behaviors of the system. The important characteristics of the signal transduction system are excitation and adaptation, and the latter implies that the transduction system can function as a "derivative sensor" with respect to the ligand concentration in that the DC component of a signal is ultimately ignored if it is not too large. This temporal sensing mechanism provides the bacterium with a memory of its passage through spatially- or temporally-varying signal fields, and adaptation is essential for successful chemotaxis. We also discuss some of the spatial patterns observed in populations and indicate how cell-level behavior can be embedded in population-level descriptions.

The Interaction of Mechanics and the Hippo Pathway in Drosophila melanogaster.

Drosophila melanogaster has emerged as an ideal system for studying the networks that control tissue development and homeostasis and, given the similarity of the pathways involved, controlled and uncontrolled growth in mammalian systems. The signaling pathways used in patterning the Drosophila wing disc are well known and result in the emergence of interaction of these pathways with the Hippo signaling pathway, which plays a central role in controlling cell proliferation and apoptosis. Mechanical effects are another major factor in the control of growth, but far less is known about how they exert their control. Herein, we develop a mathematical model that integrates the mechanical interactions between cells, which occur via adherens and tight junctions, with the intracellular actin network and the Hippo pathway so as to better understand cell-autonomous and non-autonomous control of growth in response to mechanical forces.