In this talk, we will discuss how the ideal free distribution (IFD) arises in the context of a partially migrating population using a stage-structured matrix model. Partial migration is a unique form of phenotypic diversity wherein migrant and non-migrant individuals coexist together. We show that the ideal free distribution is evolutionary stable in a global sense, assuming that both migrants and non-migrants experience density dependent competition with each other during reproduction. We also establish that the partially migrating species satisfies a dichotomy: Either both morphs have the same fitness, a scenario that corresponds to an IFD. Or, one morph has a higher fitness than the other. Evolutionary process however, will drive the population to the IFD.

Atherosclerosis, an arterial disease that causes malfunction of the cardiovascular system, occurs due to the accumulation of plaque in the intima, the innermost layer of artery. A suitable mathematical model is presented here in terms of a nonlinear autonomous system of ten Ordinary differential equations that incorporate various cellular components such as low-density lipoproteins (LDL) high-density lipoproteins (HDL), free radicals, oxidized LDL, chemoattractant, monocytes, macrophages, T-cells, smooth muscle cells (SMCs) and necrotic core (or plaque cells) as dependent variables. The present model is found to be globally stable theoretically and numerically under certain conditions. Since the model system is large in size, the concept of global stability can be physically visualized through appropriate projections of specific subsystems into three- and two-dimensional subspaces. Since the present model is globally stable, it can resist to some extent any wider arbitrary range of assumed parameter values not found in the literature. The aim of the model under study is to provide a computational framework that allows searching for important parameters that are likely to aid in clinical investigations of this malignant disease.

Agent based modeling frameworks such as PhysiCell contain many discrete and continuum components such as the cells and tissue microenvironment. These components events require use of uncoupled solvers to evaluate the problem across simulation time. At the interface between solvers special cases must be defined as low concentrations can become a source of error where fractions of objects are consumed by many discrete cells- similarly when a discrete (Cell) object is translated to a continuum framework (ODE) attention must be also payed to reduce error in the event. Better reduction of error allows for larger simulations for a cheaper cost, such as if an epithelium model (spatially simulated) is coupled to the lymph node (ODE). Examples with an infection model of COVID are shown including lymph node recruitment of immune cells.

we will consider a group of individuals, who each have a target number of contacts they would like to have with other group members. We are interested in how close this can some to being realized, and consider the long term outcome for the group under a reasonable dynamics on the number of contacts. The individuals will be represented as vertices, and the number of contacts as the vertex degree. It is well known that not all degree sequences can be realized as undirected graphs and the Havel-Hakimi algorithm characterizes those that can. Our main concern is to reach graphs that minimize the total deviation between what is desired and what is possible. The set of all such graphs and the set of all such associated sequences are termed the minimal sets. This problem has previously been considered by Broom and Cannings, and it is a hard problem to tackle for general target sequences. We consider the n-element arithmetic sequence for general n, including obtaining a formula which generates the size of the minimal set for a given arithmetic sequence. We also consider a strategic version of the model where the individuals are involved in a multiplayer game.

We devise a theoretical framework and a numerical method to infer trajectories of a stochastic process from snapshots of its temporal marginals. This problem arises in the analysis of single cell RNA-sequencing data, which provide high dimensional measurements of cell states but cannot track the trajectories of the cells over time. We prove that for a class of stochastic processes it is possible to recover the ground truth trajectories from limited samples of the temporal marginals at each time-point, and provide an efficient algorithm to do so in practice. The method we develop, Global Waddington-OT (gWOT), boils down to a smooth convex optimization problem posed globally over all time-points involving entropy-regularized optimal transport. We demonstrate that this problem can be solved efficiently in practice and yields good reconstructions, as we show on several synthetic and real datasets.

A key challenge in systems biology is creating mathematical models that can be easily and accurately combined with other models. Such models will need to share a consistent modelling framework and be easily reusable by systems biologists.One solution to this challenge is to use a physics-based approach to modelling. Bond graphs are an energy-based modelling framework that describe the rate of energy flow (power) moving through system components. By construction, bond graphs models enforce physical and thermodynamic constraints, making model components physically consistent with one another. Bond graphs also provide a graphical representation of the model and allow for easy hierarchical modelling.To demonstrate bond graph modelling applied to biological systems, we have applied this framework to Two Component Systems (TCS). TCS are a signalling mechanism found in many common bacteria such as E. coli and B. subtilis. By modelling the explicit energy dependence of TCS using bond graphs, we find new insights into the behaviour of the system in different energy contexts. A modular framework also means we can combine models together to investigate coupling dynamics of TCS. In future, we argue that such an approach could lead towards the development of a systems-wide, physically plausible whole-cell model.

Neurodegenerative disease (ND) is an umbrella term used to classify medical conditions associated with neuronal atrophy and gradual loss of cognitive abilities. The most common ND is Alzheimer's disease (AD). However, the approved drugs mostly treat the symptoms of AD. Therapeutic approaches targeting Amyloid beta (Aβ) aggregation fail to reverse or inhibit disease progression. These observations point towards gaps in the understanding of disease mechanisms. During development, the progenitor cells mature into neurons and they switch to a post mitotic, resting state. However, cell cycle reentry often precedes neuronal apoptosis hinting at a close interaction between the two processes. In this study we develop mathematical models of multiple pathways leading to cell cycle re-entry in neurons. These models incorporate the cross talk between cell cycle, neuronal and apoptotic signaling mechanisms. Our study shows that different self-sustaining feedback loops operate in post mitotic neurons that can make the cell cycle re-entry and transition to an apoptotic state irreversible. Important cell cycle regulators that function as hub nodes were identified. Further, we propose a combinatorial therapy targeting Aβ proteolysis as well as blocking the cell cycle feedback loop may alleviate the severity of the disease.

Colorectal Cancer is one of the most prevalent forms of cancer within western society. It is known to develop within the epithelia of the colon, localised to distinct invaginations within the intestinal wall, known as the crypts of Lieberkürn. While much is known about these crypts, the biomechanical process responsible for their structural maintenance remains unknown. One such process believed to be responsible for the crypts structural stability is believed to be a result of the surrounding stromal tissue.Here, we will present a 3D, multilayer, cell-centre model of tissue deformation, where cell movement is governed by the minimisation of a bending potential across the epithelium, cell-cell adhesion, and viscous effects. Using this model, we will show how the tissue is capable of maintaining a consistent structure while undergoing self renewal. We will also show how the model extends natural to describe general tissue deformations, and we hope to further extend it to describe crypt dynamic homeostasis.

A popular measure of differentiation in biodiversity is the Bray Curtis index of dissimilarity. It has recently also been proposed for use in molecular ecology. However, this measure currently cannot be predicted under specified conditions of population size, dispersal and speciation or mutation. Here I show forecasts for Bray-Curtis for two-variant systems such as single-nucleotide polymorphisms (SNPs) (or two species ecosystems). These are derived from well-known equations for population genetics, and shown to be appropriate by simulation. Thus, Bray-Curtis can now be used for assessment of differentiation, in order to understand natural or artificial processes, in addition to other measures such as Morisita-Horn/D_EST, G_ST and Shannon Mutual Information/Shannon Differentiation.

Over the last decade, several studies have discussed the importance of individual and trait variation in natural populations. However, trait variations are typically ignored in many theoretical studies of population dynamics, including those of bistable systems. In this talk, I present an analysis of a mean-field model of savanna-forest bistable dynamics -- modified from Staver and Levin, 2012, Am Nat to incorporate trait variation. Model parameters usually depend on trait values and distribution and are coupled to the state-variable in two ways: (i) as a coefficient to a state-variable or (ii) as a nonlinear function in which the trait variable and the state-variable cannot be separated as a product. Our model predicts that, in the first case, trait variation does not qualitatively affect the dynamics of the system, whereas, in the second case, it may change the dynamics. Within our model, we show that trait variation in the parameter sapling-to-adult recruitment rate affects bistability. We find that an increase in this trait variation shrinks the bistability region, or conversely, low trait variation allows the coexistence of the two stable states. Thus, we argue trait variation has important implications for the stability of ecosystems.

Interregional migration, as well as fertility and mortality, are essential ingredients of population dynamics. The general Leslie matrix is an essential tool in expressing a multi-regional population model and has been studied in demography since the 1970s. On the other hand, to study each effect of matrix entries on the eigenvalue, sensitivity analysis has also developed in ecology since the same age. Those two methodologies associate with each other via the eigensystem of the matrix model. This study reconstructs the eigensystem in the general Leslie matrix model from the perspective of the statistics for interregional migration pathways over a generation. Using this reconstructed eigensystem, we provide the sensitivity analysis consisting of the statistics of the interregional pathways. As an application of our framework, we use the latest data in Japan's declining birthrate for more than 40 years and clarify the interregional migration and the regional fertility rate that most influence population decline.

In this study, we aim to introduce new models of purposeful kinesis with diffusion coefficient dependent on fitness. New models include one additional parameter, intensity of kinesis, and may be considered as the minimal models of purposeful kinesis. It is demonstrated how kinesis could be beneficial for assimilation of patches of food or periodic fluctuations. Nevertheless, kinesis is not always beneficial in the long-term and spatially global perspective: for example, for species with the Allee effect it can delay invasion and spreading. We will also present the impact of purposeful kinesis on travelling waves. Both monotonic and non-monotonic (Allee effect) dependence of the reproduction coefficient on the population density will be presented. The possible benefits of the purposeful kinesis are demonstrated: with the higher diffusion, while the population without kinesis ends up with extinction, with kinesis stays alive and has the travelling wave behaviour. While the kinesis of the prey population is decreasing, the wave amplitude gets smaller. On the other hand, for the lower kinesis of predators, they have a sharp increase.

Multicellular systems biology is an interdisciplinary field that attracts researchers with diverse technical expertise. Simulation software are basic tools to work in multicellular systems biology. Understanding these simulation software entails a steep learning curve for interdisciplinary researchers joining multicellular systems biology research. Thus, there is a need for interactive training materials for those researchers. We have created eight cloud- hosted, interactive apps to train new users on using PhysiCell, a physics-based multicellular simulation software. These applications are open source and hosted on nanoHUB, an open and free platform to host computational simulations. We have created apps for exploring different modules in PhysiCell such as chemical diffusion, cell motility etc. In each app we fixed all parameters except those regulating one module This divide and conquer approach helps users to focus on the function of one module only. For each module user can change the parameters controlling that one aspect of simulation and run it in their browser to observe its effect. These apps can be used for training for new users as well as parameter tuning for more intermediate level users. We also added a detailed simulation where users can change all modules together to study interaction between them.

Students often question the relevance and inclusion of Calculus subject in senior high school even if their career paths lead to Biology, medicine and allied health sciences. This query stems from the engineering-directed learning materials and the non-inclusive treatment of Calculus in the classroom. To address this polarity, biology-concept embedded learning modules and lesson exemplars were developed. Using quasi-experimental design, students' attitude towards Applied Mathematics and Calculus and performance were appraised. Results indicated that integrating biology concepts and problems in calculus classes improved both students' attitudes and performance in Calculus. Implication of the study shall be discussed in terms of students' acquisition on the interface between mathematical and biological sciences.

Cholera is a severe diarrheal infection caused by the Vibrio cholerae bacterium. It affects millions of people globally with an estimated 2.9 million cases reported annually. In this study, we formulate a multi-scale model linking the between-host and within-host dynamics of cholera to gain new perspectives on the spread of the infection. We conduct a time-scale analysis for the within-host system, where the dynamics of the immune response and the pathogen are differentiated using time scales. The approach used allows for the elimination of the pathogen after a defined time, which is contrary to other within-host models. We use the within-host system as a basis for the formulation of the epidemic model which takes into account direct human-to-human transmission as well as transmission via the environment. The epidemic model is a physiologically structured model based on the immune status, which is a function derived from the within-host immune response. The basic reproduction number is derived and the steady states analysed. Analysis of the endemic equilibrium reveals conditions that may lead to its stability as well as its destabilisation through the occurrence of a Hopf bifurcation. Without loss of immunity, the environmental transmission route is necessary for periodic orbits to occur.

The development of drug resistance remains a major challenge in the treatment of chronic infections as there are little to no new drugs being discovered and mortality due to such infections are on the increase. To address this, the concept of collateral sensitivity cycling was proposed as a plausible therapy scheduling approach whereby drugs are sequentially used based on their collateral susceptibility profiles. Using control engineering approaches, we develop strategies aimed at minimizing the appearance of drug-resistant pathogens within the host whilst considering their collateral susceptibilities. With a generalized mathematical model based on bacteria population, we develop switching drug strategies which can be used to ensure the stability of the eradication equilibrium. Our numerical simulations compare different switching drug strategies and validate their use for mitigation against bacterial resistance.

Treatment with the drug 1-methyltryptophan (1-MT) has been shown to modulate immune responses by targeting several immunologically relevant pathways. One potential mode of action of 1-MT is a shift towards production of the tryptophan (TRP) metabolite kynurenic acid (KYNA), which mediates crucial immunomodulatory effects under inflammatory conditions. It is still unknown whether 1-MT is metabolized to KYNA directly or via intermediate metabolites such as TRP or kynurenine (KYN). To answer this question, this study employs mathematical modeling to evaluate six different hypothetical mechanisms. We developed models based on system of ordinary differential equations, and compare simulations to data measured in an experiment pig model investigating in vivo effects of 1-MT. We in silico evaluate the feasibility of assumptions made by comparing model dynamics with kinetic experimental data, and are thus able to direct further experimental work to the most promising explanatory mechanisms, facilitating further experimental verification. Based on analysis of the computational model, we conclude that a direct degradation of 1-MT to KYNA is the most probable metabolic pathway which best explains the experimentally observed kinetics.

The heterogeneity in vaccine coverage both locally and worldwide represents a large potential for SARS-CoV-2 variants that escape existing immunity both from vaccines and prior infections. As countries with high vaccine coverage prepare to relax other control measures, such variants could have a devastating effect. We demonstrate that, even when the variant is less transmissible than the locally dominant variant, reduced immunity can lead to a significant wave of infection. We use an SEIR ODE model of infection with two variants and three potential vaccines, and assume asymmetric immunity granted by prior infection between the two variants. We apply our model to the context in the UK, which had given first doses to 57% of its population by 29th March, with a combination of AstraZeneca and Pfizer vaccines. Initial doses focussed on older age groups and doses were given 12 weeks apart. As of 29th March, cases were dominated by the B.1.1.7 (UK) variant and the primary concern was importations of the B.1.351 (South African) variant. We show that if the B.1.351 variant significantly evades vaccine-derived and prior immunity then planned relaxations could lead to a large wave of B.1.351 infections.

Reaction-diffusion equations (RDEs) are often derived as continuum limits of lattice-based discrete models. Recently, a discrete model which allows the rates of movement, proliferation and death to depend upon whether the agents are isolated has been proposed, and this approach gives various RDEs where the diffusion term is convex and can become negative (Johnston et al., Sci. Rep. 7, 2017). Numerical simulations suggest these RDEs, under certain choices of the system parameters, support smooth and shock-fronted travelling waves. In this talk, I will formalise these preliminary numerical observations by analysing these two types of travelling wave solutions through a dynamical systems approach.

In this study, we develop a self-propelled particle model of pedestrian motion based on social interaction forces for passengers boarding and disembarking from buses in the Philippines. The forces—attractive, repulsive, and frictional—are derived from positional data acquired by tracking numerous drone videos of a section of a busy highway in Metro Manila. Model validity is tested by simulating passenger trajectories and comparing these simulations with actual data. The model will be used to explore the transmission of COVID-19 through the Philippine public transportation system and to build a framework with which policy interventions and organized crowd control systems (e.g., proper queuing systems and bus stop usage, physical distancing requirements to prevent disease transmission) can be conceptualized and justified. In the future, we will be analyzing the spread of COVID-19 by introducing a variable number of infected passengers to the system and use existing transmission onset data to give particular attention to pre-symptomatic transmission. While the study focuses on bus passengers, the model can be easily modified to be applied to other modes of transportation. In addition, while the study is motivated by COVID-19, the framework to be generated will be usable for analyzing outbreaks of other infectious diseases.

It is becoming obvious that we need better modelling tools as we strive to build larger and more interpretable models of biological systems. Specifically, we cannot keep relying on bespoke, hand-coded models when we want to be able to explore model spaces comprehensively.In this talk I demonstrate that using hypergraphs results in a computationally efficient scheme to represent and generate mathematical models of biological systems. I show that chemical hypergraphs can represent biochemical reaction networks exactly, and that hypergraphs have the potential to represent dynamical systems more generally. This framework allows us to easily manipulate and compose mathematical models, even of large systems, thereby enabling us to explore model spaces automatically. I illustrate the use of chemical hypergraphs using models of gene regulation processes.This framework allows us to construct models that other approaches may be unable to -- considering feedback loops, for instance, is trivial here. Further advantages of this approach are a more accessible, less error-prone and reproducible modelling process.

Networks can represent entities as disparate as genes, computers, infected people, predators and prey or neurons of the brain. Details of underlying structures for given systems amenable to network representations are typically limited to numbers of connections between entities or their node-degree. These degrees may be number of sexual partners, prey species, or synaptic connections in a brain. Realising a network with a given sequence of node-degrees presents a challenge especially if multiple connections or loop-backs for nodes are forbidden — otherwise known as a simple graph. Standard methods of network assembly for sampling a graph space, or all potential realisations of some degree sequence, typically require significant post-processing of initial assemblies to remove multiple connections and loop-backs. We devised an alternative method that not only permits outright exclusion of these edges, but also can target prescribed proportions of them for networks with weighted-edges. These weights may represent multiple interactions, say, between sexual partners, prey species or synaptic connections. We present our method that successfully builds networks with order 10^7 edges on scales of minutes running on a laptop, as well as links to our implementation on the GitHub repository.

In this talk I will develop a mathematical model of of the Type VI bacterial secretion system. The Type VI Secretion System (T6SS) is a transmembrane macro-molecular contractile machine able to dynamically and repeatedly compete against both prokaryotes and eukaryotes. Whilst many of the core molecular components of the T6SS have been identified, there are open questions regarding how the T6SS is regulated in response to stimuli. Here I develop a stochastic differential equation model that describes post-translational regulation of the T6SS. The model is solved numerically and used to explore how the time between successive firing events allows for spatial reorientation of firing and thus for a bacterium to respond to spatially localised external stimuli.

Planar Cell Polarity (PCP) is an evolutionary conserved phenomenon in which cells in an epithelium are polarized within the plane of the tissue. Disruptions in PCP can often lead to developmental abnormalities such as defects in neural tube formation and atypical organ shape. The emergent multi-scale dynamics of PCP has been investigated experimentally and computationally through focusing on different modules (set of molecular players) revealing asymmetric protein localisation on cell boundaries as the primary PCP mechanism. Additional ingredient in the PCP establishment is the global cue in the form of tissue-wide protein concentrations. Despite multiple mechanistic modelling attempts, an analytic understanding of the system has not yet been comprehensively achieved. Here, we present a minimal continuum model, derived from the microscopic interactions of the proteins, to study the emergence of macroscopic tissue-wide polarity. We obtain necessary and sufficient conditions and diverse parameter regimes for different cases of establishing PCP and its disruptions. We also solve the model numerically to study the model where analytic solution is not possible. Finally, we compare the model results to other existing mechanistic approaches to draw conceptual parallels between them, with the goal of identifying design principles of PCP.

Diverse biological processes like embryogenesis, morphogenesis, neurogenesis, regeneration, wound healing, and disease propagation like cancer-metastasis involve numerous cells exhibiting coherent migration. Multicellular clusters undergo dynamic rearrangement while relocating — bigger clusters split, smaller sub-clusters collide and reassemble, gaps continually appear and disappear, and cells (and the clusters as a whole) undergo variations in shape and orientation. The connections between cell-level adhesion and cluster-level dynamics, as well as the resulting consequences for cluster properties such as migration velocity, remain poorly understood. To unveil the underlying mechanics of collective migration of two-dimensional cell clusters concertedly tracking chemical gradients, we develop a generic computational framework based on the cellular Potts model which captures cell shape changes and cluster rearrangement. We find that cells have an optimal adhesion strength that maximizes cluster migration speed. The optimum negotiates a tradeoff between preserving cell-cell contact and maintaining configurational freedom, and we identify maximal variability in the cluster aspect ratio as a revealing signature. Our results suggest a collective benefit for intermediate cell-cell adhesion.U. Roy and A. Mugler. Phys. Rev. E 103, 032410 (2021).

The size distributions of persons' names, i.e., how many people share a certain name, obeys Zipf's law in various countries or areas. There is, however, a regionality for each country or area, and the ingredients of the distributions differ from each other. We statistically analyze the distributions of persons' family and given names obtained from a telephone directory to characterize their heterogeneities. By using Kullback–Leibler divergence, the inhomogeneity of name distribution are analyzed both from the viewpoint of person and prefecture.

Survival of living cells underlies many influences such as nutrient saturation, oxygen level, drug concentrations or mechanical forces. Data-supported mathematical modeling can be a powerful tool to get a better understanding of cell behavior in different settings. However, under consideration of numerous environmental factors mathematical modeling can get challenging. We present an approach to model the separate influences of each environmental quantity on the cells in a collective manner by introducing the 'environmental stress level'. It is an artificial, immeasurable variable, which quantifies to what extent viable cells would get in a stressed state, if exposed to certain conditions. A high stress level can inhibit cell growth, promote cell death and influence cell movement. As a proof of concept, we compare two systems of ordinary differential equations, which model tumor cell dynamics under various nutrient saturations respectively with and without considering an environmental stress level. Particle-based Bayesian inversion methods are used to calibrate unknown model parameters with time resolved measurements of in vitro populations of liver cancer cells. While predictions of both calibrated models show good agreement with the data, the model considering the stress level yields a better fitting.

Phenotypic (non-genetic) heterogeneity has significant implications for development and evolution of organs, organisms, and populations. Recent observations in multiple cancers have unravelled the role of phenotypic heterogeneity in driving metastasis and therapy recalcitrance. However, the origins of such phenotypic heterogeneity are poorly understood in most cancers. Here, we investigate a regulatory network underlying phenotypic heterogeneity in small cell lung cancer, a devastating disease with no molecular targeted therapy. Discrete and continuous dynamical simulations of this network reveal its multistable behavior that can explain co-existence of four experimentally observed phenotypes. Analysis of the network topology uncovers that multistability emerges from two teams of players that mutually inhibit each other but members of a team activate one another, forming a 'toggle switch' between the two teams. Deciphering these topological signatures in cancer-related regulatory networks can unravel their 'latent' design principles and offer a rational approach to characterize phenotypic heterogeneity in a tumor.

We present a model of glioblastoma (GBM) invasion which includes mass effects and tissue mechanics. Furthermore, we show how different brain tissue elasticity models affect the dynamics and invasion wave speed. Inspired by Budday et al. (2017) who mechanically tested brain tissue to determine an appropriate constitutive model of brain tissue mechanics, we explore two models: The linear elasticity model, and the one-term Ogden model. In a simplified 1D version of the model, we show the existence of travelling wave solutions. The traveling waves can be viewed as the invasion of GBM tumor cells into the surrounding healthy brain tissue. Thus, identifying the speed of the wave and how it is affected by model components and parameters is useful in determining what drives invasion. We show that although the wave speed is independent of the chosen mechanical model, the dynamics of GBM spread and the effects on surrounding brain tissue differ significantly between the linear and one-term Ogden elasticity models. Simulations predict opposite modes of GBM invasion depending on the mechanical model, with the linear and one-term Ogden models showing that GBM invades via either “pushing” or “pulling” on the surrounding tissue, respectively.

The intestinal epithelium is one of the fastest renewing tissues in mammals and shows a remarkable degree of stability towards external perturbations such as physical injuries or radiation damage. This process is driven by intestinal stem cells as well as by differentiated cells being able to revert back to a stem cell state in situations of tissue regeneration. Self-renewal and regeneration, however, require a tightly regulated balance to uphold tissue homoeostasis, as failures in maintaining this balance may lead to tissue extinction or to unbounded growth, thereby giving rise to cancerous lesions.Here, we present and analyze a mathematical model of intestinal epithelium population dynamics. The model allows to derive conditions for stability and thereby helps to identify mechanisms that lead to loss of homoeostasis, causing either regenerative failure or unbounded, malignant growth. One of the key results is the existence of specific thresholds in feedbacks after which unbounded growth occur, and a subsequent convergence of the system to a stable ratio of stem to non-stem cells. Additionally, we demonstrate how allowing for dedifferentiation enables the system to recover more gracefully after certain external perturbations from equilibrium, however opens up another way to malignant growth.