Modeling genome evolution with a diffusion approximation of a birth-and-death process

MOTIVATION: In our previous studies, we developed discrete-space birth, death and innovation models (BDIMs) of genome evolution. These models explain the origin of the characteristic Pareto distribution of paralogous gene family sizes in genomes, and model parameters that provide for the evolution of these distributions within a realistic time frame have been identified. However, extracting the temporal dynamics of genome evolution from discrete-space BDIM was not technically feasible. We were interested in obtaining dynamic portraits of the genome evolution process by developing a diffusion approximation of BDIM. RESULTS: The diffusion version of BDIM belongs to a class of continuous-state models whose dynamics is described by the Fokker-Plank equation and the stationary solution could be any specified Pareto function. The diffusion models have time-dependent solutions of a special kind, namely, generalized self-similar solutions, which describe the transition from one stationary distribution of the system to another; this provides for the possibility of examining the temporal dynamics of genome evolution. Analysis of the generalized self-similar solutions of the diffusion BDIM reveals a biphasic curve of genome growth in which the initial, relatively short, self-accelerating phase is followed by a prolonged phase of slow deceleration. This evolutionary dynamics was observed both when genome growth started from zero and proceeded via innovation (a potential model of primordial evolution), and when evolution proceeded from one stationary state to another. In biological terms, this regime of evolution can be tentatively interpreted as a punctuated-equilibrium-like phenomenon whereby evolutionary transitions are accompanied by rapid gene amplification and innovation, followed by slow relaxation to a new stationary state.     

Carrier facilitated diffusion

The concept of a mobile carrier combining reversibly with a substrate is considered as a possible mechanism for facilitated transport across biological membranes. The mathematical model is a system of three reaction diffusion equations with certain boundary conditions. Two limiting cases are discussed in detail: The case of a "thin" membrane where the diffusion of bound and unbound carrier from one surface to the other may be simulated by a single jump. If the diffusion rate of the substance to be transported is small, then an approximate stationary solution is derived using singular perturbation theory. Finally, the results of numerical simulations are presented for a wide range of parameters.     

Conditions for the existence of stationary densities for some two-dimensional diffusion processes with applications in population biology

Conditions are derived that we conjecture are necessary and sufficient for the existence of stationary densities for a class of two-dimensional diffusion processes. The derivation of the conditions rests on the assumption that a two-dimensional stationary density (which can be viewed as a stable “internal equilibrium”) exists if and only if all “boundary equilibria” are unstable in the sense that small perturbations lead to moving away from the boundaries with high probability. For the models considered, the boundary equilibria are one-dimensional stationary densities and equilibrium points. To demonstrate the usefulness of the conditions, three random environment models are analyzed: a three-allele selection model, a two-species competition model, and a two-locus selection model. Several of the results obtained have been verified by alternate methods.     

Modeling and testing treated tumor growth using cubic smoothing splines

Human tumor xenograft models are often used in preclinical study to evaluate the therapeutic efficacy of a certain compound or a combination of certain compounds. In a typical human tumor xenograft model, human carcinoma cells are implanted to subjects such as severe combined immunodeficient (SCID) mice. Treatment with test compounds is initiated after tumor nodule has appeared, and continued for a certain time period. Tumor volumes are measured over the duration of the experiment. It is well known that untreated tumor growth may follow certain patterns, which can be described by certain mathematical models. However, the growth patterns of the treated tumors with multiple treatment episodes are quite complex, and the usage of parametric models is limited. We propose using cubic smoothing splines to describe tumor growth for each treatment group and for each subject, respectively. The proposed smoothing splines are quite flexible in modeling different growth patterns. In addition, using this procedure, we can obtain tumor growth and growth rate over time for each treatment group and for each subject, and examine whether tumor growth follows certain growth pattern. To examine the overall treatment effect and group differences, the scaled chi-squared test statistics based on the fitted group-level growth curves are proposed. A case study is provided to illustrate the application of this method, and simulations are carried out to examine the performances of the scaled chi-squared tests.     

Spatio-temporal pattern formation in Rosenzweig–MacArthur model: Effect of nonlocal interactions

Spatio-temporal pattern formation in reaction–diffusion models of interacting populations is an active area of research due to various ecological aspects. Instability of homogeneous steady-states can lead to various types of patterns, which can be classified as stationary, periodic, quasi-periodic, chaotic, etc. The reaction–diffusion model with Rosenzweig–MacArthur type reaction kinetics for prey–predator type interaction is unable to produce Turing patterns but some non-Turing patterns can be observed for it. This scenario changes if we incorporate non-local interactions in the model. The main objective of the present work is to reveal possible patterns generated by the reaction–diffusion model with Rosenzweig–MacArthur type prey–predator interaction and non-local consumption of resources by the prey species. We are interested in the existence of Turing patterns in this model and in the effect of the non-local interaction on the periodic travelling wave and spatio-temporal chaotic patterns. Global bifurcation diagrams are constructed to describe the transition from one pattern to another one.     

Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations

We investigate the emergence of spatio-temporal patterns in ecological systems. In particular, we study a generalized predator-prey system on a spatial domain. On this domain diffusion is considered as the principal process of motion. We derive the conditions for Hopf and Turing instabilities without specifying the predator-prey functional responses and discuss their biological implications. Furthermore, we identify the codimension-2 Turing-Hopf bifurcation and the codimension-3 Turing-Takens-Bogdanov bifurcation. These bifurcations give rise to complex pattern formation processes in their neighborhood. Our theoretical findings are illustrated with a specific model. In simulations a large variety of different types of long-term behavior, including homogenous distributions, stationary spatial patterns and complex spatio-temporal patterns, are observed.     

Determination of diffusion coefficients of proteins in stationary phases by frontal chromatography

A strategy to determine effective diffusion coefficients of proteins in chromatographic gels is presented in this article. An experimental methodology based on frontal liquid chromatography was combined with a numerical methodology based on a mathematical model describing the chromatographic process including the extra-column dispersion, the dispersion due to the packed bed, the external mass transfer from the bulk phase to the stationary phase, and the diffusive transport within the stationary phase. The methodology has several advantages compared to previously reported methods to determine diffusion coefficients in that no other equipment than an HPLC is required, any class of stationary phases can be investigated as long as the experiments are performed under non-binding conditions, and no modification, e.g., moulding of slabs or membranes, to the stationary phase is required. To show the applicability of the methodology, the effective diffusion coefficients of lysozyme, bovine serum albumin, and immunoglobulin gamma in Sepharose CL-4B were determined and shown to be comparable with those determined with other methods.     

A spatio-temporal model of Notch signalling in the zebrafish segmentation clock: conditions for synchronised oscillatory dynamics

In the vertebrate embryo, tissue blocks called somites are laid down in head-to-tail succession, a process known as somitogenesis. Research into somitogenesis has been both experimental and mathematical. For zebrafish, there is experimental evidence for oscillatory gene expression in cells in the presomitic mesoderm (PSM) as well as evidence that Notch signalling synchronises the oscillations in neighbouring PSM cells. A biological mechanism has previously been proposed to explain these phenomena. Here we have converted this mechanism into a mathematical model of partial differential equations in which the nuclear and cytoplasmic diffusion of protein and mRNA molecules is explicitly considered. By performing simulations, we have found ranges of values for the model parameters (such as diffusion and degradation rates) that yield oscillatory dynamics within PSM cells and that enable Notch signalling to synchronise the oscillations in two touching cells. Our model contains a Hill coefficient that measures the co-operativity between two proteins (Her1, Her7) and three genes (her1, her7, deltaC) which they inhibit. This coefficient appears to be bounded below by the requirement for oscillations in individual cells and bounded above by the requirement for synchronisation. Consistent with experimental data and a previous spatially non-explicit mathematical model, we have found that signalling can increase the average level of Her1 protein. Biological pattern formation would be impossible without a certain robustness to variety in cell shape and size; our results possess such robustness. Our spatially-explicit modelling approach, together with new imaging technologies that can measure intracellular protein diffusion rates, is likely to yield significant new insight into somitogenesis and other biological processes.     

A Model of Spatial Epidemic Spread When Individuals Move Within Overlapping Home Ranges

One of the central goals of mathematical epidemiology is to predict disease transmission patterns in populations. Two models are commonly used to predict spatial spread of a disease. The first is the distributed-contacts model, often described by a contact distribution among stationary individuals. The second is the distributed-infectives model, often described by the diffusion of infected individuals. However, neither approach is ideal when individuals move within home ranges. This paper presents a unified modeling hypothesis, called the restricted-movement model. We use this model to predict spatial spread in settings where infected individuals move within overlapping home ranges. Using mathematical and computational approaches, we show that our restricted-movement model has three limits: the distributed-contacts model, the distributed-infectives model, and a third, less studied advective distributed-infectives limit. We also calculate approximate upper bounds for the rates of an epidemic's spatial spread. Guidelines are suggested for determining which limit is most appropriate for a specific disease.     

Homogenization Theory for the Prediction of Obstructed Solute Diffusivity in Macromolecular Solutions

The study of diffusion in macromolecular solutions is important in many biomedical applications such as separations, drug delivery, and cell encapsulation, and key for many biological processes such as protein assembly and interstitial transport. Not surprisingly, multiple models for the a-priori prediction of diffusion in macromolecular environments have been proposed. However, most models include parameters that are not readily measurable, are specific to the polymer-solute-solvent system, or are fitted and do not have a physical meaning. Here, for the first time, we develop a homogenization theory framework for the prediction of effective solute diffusivity in macromolecular environments based on physical parameters that are easily measurable and not specific to the macromolecule-solute-solvent system. Homogenization theory is useful for situations where knowledge of fine-scale parameters is used to predict bulk system behavior. As a first approximation, we focus on a model where the solute is subjected to obstructed diffusion via stationary spherical obstacles. We find that the homogenization theory results agree well with computationally more expensive Monte Carlo simulations. Moreover, the homogenization theory agrees with effective diffusivities of a solute in dilute and semi-dilute polymer solutions measured using fluorescence correlation spectroscopy. Lastly, we provide a mathematical formula for the effective diffusivity in terms of a non-dimensional and easily measurable geometric system parameter.     

Quaternary range and demographic expansion of Liolaemus darwinii (Squamata: Liolaemidae) in the Monte Desert of Central Argentina using Bayesian phylogeography and ecological niche modelling

Until recently, most phylogeographic approaches have been unable to distinguish between demographic and range expansion processes, making it difficult to test for the possibility of range expansion without population growth and vice versa. In this study, we applied a Bayesian phylogeographic approach to reconstruct both demographic and range expansion in the lizard Liolaemus darwinii of the Monte Desert in Central Argentina, during the Late Quaternary. Based on analysis of 14 anonymous nuclear loci and the cytochrome b mitochondrial DNA gene, we detected signals of demographic expansion starting at ~55 ka based on Bayesian Skyline and Skyride Plots. In contrast, Bayesian relaxed models of spatial diffusion suggested that range expansion occurred only between ~95 and 55 ka, and more recently, diffusion rates were very low during demographic expansion. The possibility of population growth without substantial range expansion could account for the shared patterns of demographic expansion during the Last Glacial Maxima (OIS 2 and 4) in fish, small mammals and other lizards of the Monte Desert. We found substantial variation in diffusion rates over time, and very high rates during the range expansion phase, consistent with a rapidly advancing expansion front towards the southeast shown by palaeo‐distribution models. Furthermore, the estimated diffusion rates are congruent with observed dispersal rates of lizards in field conditions and therefore provide additional confidence to the temporal scale of inferred phylogeographic patterns. Our study highlights how the integration of phylogeography with palaeo‐distribution models can shed light on both demographic and range expansion processes and their potential causes.     

On the Modelling of Biological Patterns with Mechanochemical Models: Insights from Analysis and Computation

The diversity of biological form is generated by a relatively small number of underlying mechanisms. Consequently, mathematical and computational modelling can, and does, provide insight into how cellular level interactions ultimately give rise to higher level structure. Given cells respond to mechanical stimuli, it is therefore important to consider the effects of these responses within biological self-organisation models. Here, we consider the self-organisation properties of a mechanochemical model previously developed by three of the authors in Acta Biomater. 4, 613–621 (2008), which is capable of reproducing the behaviour of a population of cells cultured on an elastic substrate in response to a variety of stimuli. In particular, we examine the conditions under which stable spatial patterns can emerge with this model, focusing on the influence of mechanical stimuli and the interplay of non-local phenomena. To this end, we have performed a linear stability analysis and numerical simulations based on a mixed finite element formulation, which have allowed us to study the dynamical behaviour of the system in terms of the qualitative shape of the dispersion relation. We show that the consideration of mechanotaxis, namely changes in migration speeds and directions in response to mechanical stimuli alters the conditions for pattern formation in a singular manner. Furthermore without non-local effects, responses to mechanical stimuli are observed to result in dispersion relations with positive growth rates at arbitrarily large wavenumbers, in turn yielding heterogeneity at the cellular level in model predictions. This highlights the sensitivity and necessity of non-local effects in mechanically influenced biological pattern formation models and the ultimate failure of the continuum approximation in their absence.     

Parity models of cell proliferation

Models of the cell cycle are developed and simulated for renewing and for exponentially growing populations. Type A corresponds to the author's previous model (Barrett, 1966, Barrett, 1966). Type B incorporates a resting or regenerating phase, recurring regularly every few cycles, with an endogeneous rhythm. The parity of a cell is the number of cycles since the phase occurred in an ancestral cycle. Type C incorporates asymmetric cell division. Type B leads to novel features in calculated and simulated labelled mitoses curves, since increasing amplitudes of successive waves are permitted, and accords with certain tumour data that previously seemed anomalous. Type B leads also to correlations between cycle times for different generations, in accordance with patterns reported in bacteria. Some mathematical properties of models Type A and B are noted.     

Transport rates of proteins in porous materials with known microgeometry.

Many biological and biotechnological systems involve the diffusion of macromolecules through complicated macroporous (pore size on the order of 10-100 microns) environments. In this report, we present and evaluate an experimental system for measuring the rate of protein transport in an inert, macroporous membrane. For this particular membrane system, the microgeometry was characterized in terms of distribution of pore size, position, and orientation. Although the rate of protein desorption was much less than expected based on continuum diffusion models, we demonstrate that the measured transport rates are consistent with diffusion of protein in a complex, interconnected network of water-filled pores. The porous systems exhibit transitional behavior in quantitative agreement with the behavior of percolation lattices (mean square error 7%, n = 29). Predictive mathematical models of the diffusion process were developed: these models used percolation concepts to describe pore topology, continuum models of diffusion/dissolution to describe protein movement at each single pore, and measured pore size distributions. Effective diffusion coefficients for protein transport in aqueous, constricted macropores were predicted by this technique. Predicted diffusion coefficients, based on measured and derived microstructural parameters, agree with experimentally measured diffusion coefficients within a factor of 2. This approach may be useful in the design of porous polymer systems for biological applications and for evaluating other biological systems where conduction of mass, heat, momentum, or charge occurs in a heterogeneous environment.     

Growth of Populus euramericana

Growth curves of successive leaves of Populus euramericana (Dode) Guinier 'Robusta' have been determined. With ample supply of water and nutrients the growth of a poplar shoot follows a fixed pattern: an initial logarithmic acceleration phase followed by a stationary phase in which leaves of equal size are produced at a constant rate. Analysis of growth curves of leaves enabled the growth curves of leaf primordia to be predicted. These primordial growth curves are compared to the indirectly determined growth curves of primordia by measuring the lengths of successive leaf primordia in the apex during the stationary phase of growth. The increase in length of successive leaves in the acceleration phase of growth continues for a longer period at high than at low irradiace. The relative growth rates of leaf primordia, leaves and internodes are discussed in terms of shoot growth and phyllotaxis.     

Implications of diffusion and time-varying morphogen gradients for the dynamic positioning and precision of bistable gene expression boundaries

The earliest models for how morphogen gradients guide embryonic patterning failed to account for experimental observations of temporal refinement in gene expression domains. Following theoretical and experimental work in this area, dynamic positional information has emerged as a conceptual framework to discuss how cells process spatiotemporal inputs into downstream patterns. Here, we show that diffusion determines the mathematical means by which bistable gene expression boundaries shift over time, and therefore how cells interpret positional information conferred from morphogen concentration. First, we introduce a metric for assessing reproducibility in boundary placement or precision in systems where gene products do not diffuse, but where morphogen concentrations are permitted to change in time. We show that the dynamics of the gradient affect the sensitivity of the final pattern to variation in initial conditions, with slower gradients reducing the sensitivity. Second, we allow gene products to diffuse and consider gene expression boundaries as propagating wavefronts with velocity modulated by local morphogen concentration. We harness this perspective to approximate a PDE model as an ODE that captures the position of the boundary in time, and demonstrate the approach with a preexisting model for Hunchback patterning in fruit fly embryos. We then propose a design that employs antiparallel morphogen gradients to achieve accurate boundary placement that is robust to scaling. Throughout our work we draw attention to tradeoffs among initial conditions, boundary positioning, and the relative timescales of network and gradient evolution. We conclude by suggesting that mathematical theory should serve to clarify not just our quantitative, but also our intuitive understanding of patterning processes.     

The complex dynamics of a diffusive prey–predator model with an Allee effect in prey

This paper investigates complex dynamics of a predator–prey interaction model that incorporates: (a) an Allee effect in prey; (b) the Michaelis–Menten type functional response between prey and predator; and (c) diffusion in both prey and predator. We provide rigorous mathematical results of the proposed model including: (1) the stability of non-negative constant steady states; (2) sufficient conditions that lead to Hopf/Turing bifurcations; (3) a prior estimates of positive steady states; (4) the non-existence and existence of non-constant positive steady states when the model is under zero-flux boundary condition. We also perform completed analysis of the corresponding ODE model to obtain a better understanding on effects of diffusion on the stability. Our analytical results show that the small values of the ratio of the prey's diffusion rate to the predator's diffusion rate are more likely to destabilize the system, thus generate Hopf-bifurcation and Turing instability that can lead to different spatial patterns. Through numerical simulations, we observe that our model, with or without Allee effect, can exhibit extremely rich pattern formations that include but not limit to strips, spotted patterns, symmetric patterns. In addition, the strength of Allee effects also plays an important role in generating distinct spatial patterns.