On a functional equation model of transient cell growth

Begg, Ronald; Wall, David J. N.; Wake, G. C.

doi:10.1093/imammb/dqi015

Cited by 7 publications

(4 citation statements)

References 5 publications

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“…20 The solution technique for the symmetric cell division case can be adapted to determine the general solution for the case of asymmetric cell division. We conclude that adding dispersion to the cell division problem (5) does not impact the shape and positivity the SSD solution in a substantial way (See Figure 1), though it does increase the number density of smaller cells. We also conclude that shape of the SSD solution and its positivity remain largely unaffected by the mode of cell division.…”

Section: Discussionmentioning

confidence: 83%

“…The cell growth model to is based on a model proposed by Sinko and Streifer for planarian worms. The functional PDE was studied, among others, by Hall and Wake, Hall et al, Begg et al, Metz and Diekmann, and Zaidi et al Perthame and Ryzhik established the existence of a unique eigenvalue λ and the corresponding positive eigenfunction y ( x ) towards which all solutions to converge exponentially for large time, ie,

‖ e^{- λ t} n (x, t) - < n_{0} > y (x) {false‖}_{L^{1} ({double-struckR}^{+})} \to 0,

as t → ∞ . Here,

< n_{0} > = true \int_{0}^{\infty} n_{0} false(x false) d x

is a normalization constant.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Asymmetric cell division with stochastic growth rate. Dedicated to the memory of the late Spartak Agamirzayev

Efendiev

Brunt

Zaidi

et al. 2018

Math Methods in App Sciences

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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.

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Section: Discussionmentioning

confidence: 83%

‖ e^{- λ t} n (x, t) - < n_{0} > y (x) {false‖}_{L^{1} ({double-struckR}^{+})} \to 0,

as t → ∞ . Here,

< n_{0} > = true \int_{0}^{\infty} n_{0} false(x false) d x

is a normalization constant.…”

Section: Introductionmentioning

confidence: 99%

Asymmetric cell division with stochastic growth rate. Dedicated to the memory of the late Spartak Agamirzayev

Efendiev

Brunt

Zaidi

et al. 2018

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…The model discussed in [21] was studied in a much earlier paper [6], in which the authors considered steady size distributions of solutions. Related studies by this group include [4] and papers in which this type of model is applied to tumour cells [2] and plankton [3].…”

Section: Related Literaturementioning

confidence: 99%

Estimates for Approximate Solutions to a Functional Differential Equation Model of Cell Division

Taylor

Yang

2020

ANZIAM J.

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The functional partial differential equation (FPDE) for cell division, $$ \begin{align*} &\frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t))\\ &\quad = -(b(x,t)+\mu(x,t))n(x,t)+b(\alpha x,t)\alpha n(\alpha x,t)+b(\beta x,t)\beta n(\beta x,t), \end{align*} $$ is not amenable to analytical solution techniques, despite being closely related to the first-order partial differential equation (PDE) $$ \begin{align*} \frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t)) = -(b(x,t)+\mu(x,t))n(x,t)+F(x,t), \end{align*} $$ which, with known $F(x,t)$ , can be solved by the method of characteristics. The difficulty is due to the advanced functional terms $n(\alpha x,t)$ and $n(\beta x,t)$ , where $\beta \ge 2 \ge \alpha \ge 1$ , which arise because cells of size x are created when cells of size $\alpha x$ and $\beta x$ divide. The nonnegative function, $n(x,t)$ , denotes the density of cells at time t with respect to cell size x. The functions $g(x,t)$ , $b(x,t)$ and $\mu (x,t)$ are, respectively, the growth rate, splitting rate and death rate of cells of size x. The total number of cells, $\int _{0}^{\infty }n(x,t)\,dx$ , coincides with the $L^1$ norm of n. The goal of this paper is to find estimates in $L^1$ (and, with some restrictions, $L^p$ for $p>1$ ) for a sequence of approximate solutions to the FPDE that are generated by solving the first-order PDE. Our goal is to provide a framework for the analysis and computation of such FPDEs, and we give examples of such computations at the end of the paper.

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“…A symmetric division of cells which produces equal-sized daughter cells leads to a PDE with one nonlocal term. Hall and Wake [6] formulated such a model, which was then studied by them [7], Basse et al [2], Begg et al [3] and Zaidi et al [26]. [2] Asymmetrical cell division with exponential growth 71…”

Section: Introductionmentioning

confidence: 99%

Asymmetrical Cell Division With Exponential Growth

Zaidi

Brunt

2021

ANZIAM J.

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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.

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On a functional equation model of transient cell growth

Cited by 7 publications

References 5 publications

Asymmetric cell division with stochastic growth rate. Dedicated to the memory of the late Spartak Agamirzayev

Asymmetric cell division with stochastic growth rate. Dedicated to the memory of the late Spartak Agamirzayev

Estimates for Approximate Solutions to a Functional Differential Equation Model of Cell Division

Asymmetrical Cell Division With Exponential Growth

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