We give a refined description of the dominant spectrum of a non-local operator that models growth and equal mitosis of cells. More precisely we look at the spectrum in half planes at the right hand side of the first accumulation point of eigenvalues and give criteria on the weight of weighted L^1 spaces for this spectrum to be made of explicit simple eigenvalues. The method relies on a high order long time asymptotic expansion of the solutions to the associated evolution equation obtained in [Zaidi, van Brunt, Wake, Proc. A, R. Soc. Lond., 2015] combined with a Weyl theorem taken from [Mischler, Scher, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 2016].

UDC 517.9 We consider the balanced pantograph equation (BPE) y ′ ( x ) + y ( x ) = ∑ k = 1 m p k y ( a k x ) , where a k , p k > 0 and ∑ k = 1 m p k = 1. It is known that if K = ∑ k = 1 m p k ln a k ≤ 0 then, under mild technical conditions, the BPE does not have bounded solutions that are not constant, whereas for K > 0 these solutions exist. In the present paper, we deal with a BPE of mixed type, i.e., a 1 < 1 < a m , and prove that, in this case, the BPE has a nonconstant solution y and that y ( x ) ∼ c x σ as x → ∞ , where c > 0 and σ is the unique positive root of the characteristic equation P ( s ) = 1 - ∑ k = 1 m p k a k - s = 0. We also show that y is unique (up to a multiplicative constant) among the solutions of the BPE that decay to zero as x → ∞ .

In this paper, we consider a mathematical model for cell division using a Pantograph-type nonlocal partial differential equation, accompanied by relevant initial and boundary conditions. This formulation results in a nonlocal singular eigenvalue problem. We explore the possible eigenvalues that may lead to nontrivial solutions. We then consider cells that divide once they achieve a minimum size. Our model incorporates asymmetric cell division and exponential growth. We show that, unlike the constant growth rate case, a probability density function eigenvalue can be determined explicitly. Additionally, we demonstrate that a stochastic growth rate produces eigenfunctions expressed as an infinite series of modified Bessel functions. We extend our findings to encompass a wider range of dispersion and growth rates. The implications of this work are significant for understanding the dynamics of cell populations in biological systems. The work has potential applications in cancer research and developmental biology, where cell growth and division play critical roles.

Linear inverse problems for count data arise in a myriad of settings. The latent counts lie on a fibre that is too large to enumerate in most practical problems, but inference can proceed by sampling the fibre. We examine the mixing properties of hit-and-run samplers in this context. In general convergence can be arbitrarily slow. However, there is a class of linear inverse problems for which rapid mixing for uniform fibre sampling is possible, using Markov sub-bases that are of minimum size but yet provide a sufficiently rich range of sampling directions to avoid the need for zig-zagging walks to ensure connectivity. Focussing on such problems, we study a particular class of bases that enjoy these properties under certain easily checkable conditions on the configuration matrix. We also examine the mixing properties of these bases when employing commonly used Poisson models. Our theoretical results provide practical guidance on optimizing these Markov sub-bases.

Spatiotemporal patterns are common in biological systems. For electrically coupled cells, previous studies of pattern formation have mainly used applied current as the primary bifurcation parameter. The purpose of this paper is to show that applied current is not needed to generate spatiotemporal patterns for smooth muscle cells. The patterns can be generated solely by external mechanical stimulation (transmural pressure). To do this we study a reaction-diffusion system involving the Morris–Lecar equations and observe a wide range of spatiotemporal patterns for different values of the model parameters. Some aspects of these patterns are explained via a bifurcation analysis of the system without coupling — in particular Type I and Type II excitability both occur. We show the patterns are not due to a Turing instability and that the spatially extended model exhibits spatiotemporal chaos. We also use travelling wave coordinates to analyse travelling waves.

Identities between infinite products and series can prove elusive. For example, we know that $$\displaystyle \prod _{j=1}^{\infty }\left (1-\frac {4x^{2}}{(2j-1)^{2}\pi ^{2}}\right ) =\sum _{j=0}^{\infty }(-1)^j \frac {x^{2j}}{(2j)!} $$ for all $$x\in \mathbb {R}$$ , because both the product and the series represent $$\cos x$$ . The relation, however, is not obvious from the form of the product or the series, and it was not deduced by examining partial sums or products. We used the properties of the sine and cosine function along with integration to get the infinite product.

Abstract An advanced pantograph-type partial differential equation, supplemented with initial and boundary conditions, arises in a model of asymmetric cell division. Methods for solving such problems are limited owing to functional (nonlocal) terms. The separation of variables entails an eigenvalue problem that involves a nonlocal ordinary differential equation. We discuss plausible eigenvalues that may yield nontrivial solutions to the problem for certain choices of growth and division rates of cells. We also consider the asymmetric division of cells with linear growth rate which corresponds to “exponential growth” and exponential rate of cell division, and show that the solution to the problem is a certain Dirichlet series. The distribution of the first moment of the biomass is shown to be unimodal.

For statistical linear inverse problems involving count data, inference typically requires sampling a latent variable with conditional support comprising of the lattice points in a convex polytope. Irreducibility of random walk samplers is guaranteed only if a sufficiently rich array of sampling directions is available. In principle, this can be achieved by finding a Markov basis of moves ab initio, but in practice doing so may be computationally infeasible. What is more, the use of a full Markov basis can lead to very poor mixing. It is far simpler to find a lattice basis of moves, which can be tailored to the overall geometry of the polytope. However, a single lattice basis generally does not connect all points in the polytope. In response, we propose a dynamic lattice basis sampler. This sampler can access a sufficient variety of sampling directions to guarantee irreducibility, but also prefers moves that are well aligned to the polytope geometry, hence promoting good mixing. The probability with which the sampler selects different bases can be tuned. We present an efficient algorithm for updating the lattice basis, obviating the need for repeated matrix inversion.

The theory of sequences can be combined with the familiar notion of a finite product to produce a theory of infinite products. The theory of infinite series shadows most of this material because many of the convergence questions for infinite products can be answered by results from the theory of series. The theory of infinite products nonetheless has a flavour distinct from that of series, as becomes evident once the definitions of absolute and conditional convergence of products are introduced.

Evidence from experimental studies shows that oscillations due to electro-mechanical coupling can be generated spontaneously in smooth muscle cells. Such cellular dynamics are known as pacemaker dynamics. In this article, we address pacemaker dynamics associated with the interaction of $${\text {Ca}}^{2+}$$ and $$\text {K}^+$$ fluxes in the cell membrane of a smooth muscle cell. First we reduce a pacemaker model to a two-dimensional system equivalent to the reduced Morris–Lecar model and then perform a detailed numerical bifurcation analysis of the reduced model. Existing bifurcation analyses of the Morris–Lecar model concentrate on external applied current, whereas we focus on parameters that model the response of the cell to changes in transmural pressure. We reveal a transition between Type I and Type II excitabilities with no external current required. We also compute a two-parameter bifurcation diagram and show how the transition is explained by the bifurcation structure.

The magnitude of the Gibbs free energy change of the substrate transformation that supports the growth of a microbe is decreased when the concentrations of the substrates are decreased and when the concentrations of the products of metabolism are increased. Microbes require a supply of ATP for cell maintenance and growth, and coupling the transformation of substrates to products with the formation of ATP also decreases the magnitude of the Gibbs free energy change. Here we include these three thermodynamic controllers (substrate and product concentration, and ATP formation) in a model of substrate transformation by hydrogenotrophic methanogens that results in a number of realistic behaviours. First, a threshold for substrate use emerges, below which the methanogen cannot metabolise its substrate. Under this model, microbes that capture more of the Gibbs free energy change from substrate transformation in the form of ATP have greater thresholds for their substrate, in line with observations of actual microbes. Second, an apparent saturation constant emerges that is controlled by the thermodynamics of the reaction. This increases with increasing ATP synthesis per substrate, so that methanogens that conserve more ATP grow faster at higher substrate concentrations, but are less competitive at low substrate concentrations. As a result, simply changing the ATP yield (moles of ATP per mole of substrate) results in methanogens with differing ecological strategies through thermodynamic impacts on their metabolism. Third, end-product inhibition through thermodynamic feedback can limit the growth of microbes, and those that capture more ATP per substrate are limited by smaller product concentrations than those that capture less ATP.

Simple models for size structured cell populations undergoing growth and division produce a class of functional ordinary differential equations, called pantograph equations, that describe the long time asymptotics of the cell number density. Pantograph equations arise in a number of applications outside this model and, as a result, have been studied heavily over the last five decades. In this paper we review and survey the rôle of the pantograph equation in the context of cell division. In addition, for a simple case we present a method of solution based on the Mellin transform and establish uniqueness directly from the transform equation.

A cell growth model for a size‐structured cell population with a stochastic growth rate for size and division into two daughter cells of unequal size is studied in this paper. The model entails an initial boundary value problem that involves a second‐order parabolic partial differential equation with two nonlocal terms, the presence of which is a consequence of asymmetry in the cell division. The solution techniques for solving such problems are rare due to the nonlocal terms. In this paper, we solve the initial boundary value problem for arbitrary initial distributions. We obtain a separable solution, as well as the general solution to the partial differential equation, and show that the solutions converge to the separable solution for large time. As in the symmetric division case, the dispersion term does not affect the rate of convergence to the separable solution.

In this paper we solve an initial-boundary value problem that involves a pde with a nonlocal term. The problem comes from a cell division model where the growth is assumed to be stochastic. The deterministic version of this problem yields a first-order pde; the stochastic version yields a second-order parabolic pde. There are no general methods for solving such problems even for the simplest cases owing to the nonlocal term. Although a solution method was devised for the simplest version of the first-order case, the analysis does not readily extend to the second-order case. We develop a method for solving the second-order case and obtain the exact solution in a form that allows us to study the long time asymptotic behaviour of solutions and the impact of the dispersion term. We establish the existence of a large time attracting solution towards which solutions converge exponentially in time. The dispersion term does not appear in the exponential rate of convergence.

We consider an initial–boundary value problem that involves a partial differential equation with a functional term. The problem is motivated by a cell division model for size structured cell cohorts in which growth and division occur. Although much is known about the large time asymptotic behaviour of solutions to these problems for constant growth rates, general solution techniques are rare. We analyse the case where the growth rate is linear and the division rate is a monomial, and we develop a method to determine the general solution for a general class of initial data. The large time dynamics of solutions for this case are significantly different from the constant growth rate case. We show that solutions approach a time-dependent attracting solution that is periodic in the time variable.

We give here an explicit formula for the following critical case of the growth-fragmentation equation $$\frac{\partial}{\partial t} u(t, x) + \frac{\partial}{\partial x} (gxu(t, x)) + bu(t, x) = b\alpha^2 u(t, \alpha x), \qquad u(0, x) = u\_0 (x),$$ for some constants $g > 0$, $b > 0$ and $\alpha > 1$ - the case $\alpha = 2$ being the emblematic binary fission case. We discuss the links between this formula and the asymptotic ones previously obtained in (Doumic, Escobedo, Kin. Rel. Mod., 2016), and use them to clarify how periodicity may appear asymptotically.

Existing mathematical models to estimate methane production in the rumen are based on calculation of hydrogen balances without considering the presence of methanogens. In this study, a mechanistic model of methane production is proposed that depicts the interaction between hydrogen concentration and methanogens in the rumen. Analytical results show that it meets biological expectations, namely increased fractional passage rate leads to a greater growth rate of methanogens, and a greater steady state hydrogen concentration. This model provides a basis on which to develop a more comprehensive model of methane production in the rumen that includes thermodynamics and feed fermentation pathways.