Bulletin of Mathematical Biology

Gejji, Richard; Lou, Yuan; Munther, Daniel; Peyton, Justin

doi: 10.1007/s11538-011-9662-4pmid: 21557035

We study a two species competition model in which the species have the same population dynamics but different dispersal strategies and show how these dispersal strategies evolve. We introduce a general dispersal strategy which can result in the ideal free distributions of both competing species at equilibrium and generalize the result of Averill et al. (2011). We further investigate the convergent stability of this ideal free dispersal strategy by varying random dispersal rates, advection rates, or both of these two parameters simultaneously. For monotone resource functions, our analysis reveals that among two similar dispersal strategies, selection generally prefers the strategy which is closer to the ideal free dispersal strategy. For nonmonotone resource functions, our findings suggest that there may exist some dispersal strategies which are not ideal free, but could be locally evolutionarily stable and/or convergent stable, and allow for the coexistence of more than one species.

Miao, Hongyu; Jin, Xia; Perelson, Alan; Wu, Hulin

doi: 10.1007/s11538-011-9668-ypmid: 21681605

Carboxy-fluorescein diacetate succinimidyl ester (CFSE) labeling is an important experimental tool for measuring cell responses to extracellular signals in biomedical research. However, changes of the cell cycle (e.g., time to division) corresponding to different stimulations cannot be directly characterized from data collected in CFSE-labeling experiments. A number of independent studies have developed mathematical models as well as parameter estimation methods to better understand cell cycle kinetics based on CFSE data. However, when applying different models to the same data set, notable discrepancies in parameter estimates based on different models has become an issue of great concern. It is therefore important to compare existing models and make recommendations for practical use. For this purpose, we derived the analytic form of an age-dependent multitype branching process model. We then compared the performance of different models, namely branching process, cyton, Smith–Martin, and a linear birth–death ordinary differential equation (ODE) model via simulation studies. For fairness of model comparison, simulated data sets were generated using an agent-based simulation tool which is independent of the four models that are compared. The simulation study results suggest that the branching process model significantly outperforms the other three models over a wide range of parameter values. This model was then employed to understand the proliferation pattern of CD4+ and CD8+ T cells under polyclonal stimulation.

Zhang, Lai; Lin, Zhigui; Pedersen, Michael

doi: 10.1007/s11538-011-9675-zpmid: 21769516

The physiological-structured population models assume that a fixed fraction of energy intake is utilized for individual growth and maintenance while the remaining for adult fertility. The assumption results in two concerns: energy loss for juveniles and a reproduction dilemma for adults. The dilemma results from the possibility that adults have to breed even if metabolic costs fail to be covered. We consider a size-structured population model, where standard metabolism is given top priority for utilizing energy intake and the surplus energy, if there is any, is distributed to individual growth and reproduction. Moreover, the portion of surplus energy for reproduction is size-dependent and increases monotonically with size. Using the newly developed parameter continuation, we demonstrate their disparate effects on population dynamics. Results show that the size-dependent mechanism of energy allocation primarily exerts destabilizing effects on the system but considerably promotes species coexistence, in comparison with the size-independent mechanism. We conclude that the size-dependent mechanism is, to a large extent, a dispensable component of model ingredients when ontogeny is explicitly taken into consideration.

Chen, Zhen; Ghosal, Sandip

doi: 10.1007/s11538-011-9708-7pmid: 22147104

In a previous paper (Ghosal and Chen in Bull. Math. Biol. 72:2047, 2010), it was shown that the evolution of the solute concentration in capillary electrophoresis is described by a nonlinear wave equation that reduced to Burger’s equation if the nonlinearity was weak. It was assumed that only strong electrolytes (fully dissociated) were present. In the present paper, it is shown that the same governing equation also describes the situation where the electrolytic buffer consists of a single weak acid (or base). A simple approximate formula is derived for the dimensionless peak variance which is shown to agree well with published experimental data.

Kelly, W.; Stumpf, M.

doi: 10.1007/s11538-011-9680-2pmid: 21870201

Protein interaction networks comprise thousands of individual binary links between distinct proteins. Whilst these data have attracted considerable attention and been the focus of many different studies, the networks, their structure, function, and how they change over time are still not fully known. More importantly, there is still considerable uncertainty regarding their size, and the quality of the available data continues to be questioned. Here, we employ statistical models of the experimental sampling process, in particular capture–recapture methods, in order to assess the false discovery rate and size of protein interaction networks. We uses these methods to gauge the ability of different experimental systems to find the true binary interactome. Our model allows us to obtain estimates for the size and false-discovery rate from simple considerations regarding the number of repeatedly interactions, and provides suggestions as to how we can exploit this information in order to reduce the effects of noise in such data. In particular our approach does not require a reference dataset. We estimate that approximately more than half of the true physical interactome has now been sampled in yeast.

Sun, Qiwen; Tang, Moxun; Yu, Jianshe

doi: 10.1007/s11538-011-9683-zpmid: 21870200

Gene transcription is a central cellular process and is stochastic in nature. The stochasticity has been studied in real cells and in theory, but often for the transcription activated by a single signaling pathway at steady-state. As transcription of many genes is involved with multiple pathways, we investigate how the transcription efficiency and noise is modulated by cross-talking pathways. We model gene transcription as a renewal process for which the gene can be turned on by different pathways. We determine the transcription efficiency by solving a system of differential equations, and obtain the mathematical formula of the noise strength by the Laplace transform and standard techniques in renewal theory. Our numerical examples demonstrate that cross-talking pathways are capable of inducing more cells to transcribe than the steady-state level after a short time period of signal transduction, and creating exceedingly high stationary transcription noise strength. In contrast, it is shown that one signaling pathway alone is unable to do so. Very strikingly, it is observed that the noise strength varies gradually over most values of the system parameters, but changes abruptly over a narrow range in the neighborhoods of some critical parameter values.

Fudenberg, Drew; Imhof, Lorens

doi: 10.1007/s11538-011-9687-8pmid: 21901527

Cell populations can benefit from changing phenotype when the environment changes. One mechanism for generating these changes is stochastic phenotype switching, whereby cells switch stochastically from one phenotype to another according to genetically determined rates, irrespective of the current environment, with the matching of phenotype to environment then determined by selective pressure. This mechanism has been observed in numerous contexts, but identifying the precise connection between switching rates and environmental changes remains an open problem. Here, we introduce a simple model to study the evolution of phenotype switching in a finite population subject to random environmental shocks. We compare the successes of competing genotypes with different switching rates, and analyze how the optimal switching rates depend on the frequency of environmental changes. If environmental changes are as rare as mutations, then the optimal switching rates mimic the rates of environmental changes. If the environment changes more frequently, then the optimal genotype either maximally favors fitness in the more common environment or has the maximal switching rate to each phenotype. Our results also explain why the optimum is relatively insensitive to fitness in each environment.

Layne, Lori; Dimitrova, Elena; Macauley, Matthew

doi: 10.1007/s11538-011-9692-ypmid: 22139748

We introduce the nested canalyzing depth of a function, which measures the extent to which it retains a nested canalyzing structure. We characterize the structure of functions with a given depth and compute the expected activities and sensitivities of the variables. This analysis quantifies how canalyzation leads to higher stability in Boolean networks. It generalizes the notion of nested canalyzing functions (NCFs), which are precisely the functions with maximum depth. NCFs have been proposed as gene regulatory network models, but their structure is frequently too restrictive and they are extremely sparse. We find that functions become decreasingly sensitive to input perturbations as the canalyzing depth increases, but exhibit rapidly diminishing returns in stability. Additionally, we show that as depth increases, the dynamics of networks using these functions quickly approach the critical regime, suggesting that real networks exhibit some degree of canalyzing depth, and that NCFs are not significantly better than functions of sufficient depth for many applications of the modeling and reverse engineering of biological networks.

Hu, Wen-Yong; Zhong, Wei-Rong; Wang, Feng-Hua; Li, Li; Shao, Yuan-Zhi

doi: 10.1007/s11538-011-9693-xpmid: 21972030

Based on the logistic growth law for a tumour derived from enzymatic dynamics, we address from a physical point of view the phenomena of synergism, additivity and antagonism in an avascular anti-tumour system regulated externally by dual coupling periodic interventions, and propose a theoretical model to simulate the combinational administration of chemotherapy and immunotherapy. The in silico results of our modelling approach reveal that the tumour population density of an anti-tumour system, which is subject to the combinational attack of chemotherapeutical as well as immune intervention, depends on four parameters as below: the therapy intensities D, the coupling intensity I, the coupling coherence R and the phase-shifts Φ between two combinational interventions. In relation to the intensity and nature (synergism, additivity and antagonism) of coupling as well as the phase-shift between two therapeutic interventions, the administration sequence of two periodic interventions makes a difference to the curative efficacy of an anti-tumour system. The isobologram established from our model maintains a considerable consistency with that of the well-established Loewe Additivity model (Tallarida, Pharmacology 319(1):1–7, 2006). Our study discloses the general dynamic feature of an anti-tumour system regulated by two periodic coupling interventions, and the results may serve as a supplement to previous models of drug administration in combination and provide a type of heuristic approach for preclinical pharmacokinetic investigation.

Garnier, Jimmy; Roques, Lionel; Hamel, François

doi: 10.1007/s11538-011-9694-9pmid: 21972031

We analyze the role of the spatial distribution of the initial condition in reaction–diffusion models of biological invasion. Our study shows that, in the presence of an Allee effect, the precise shape of the initial (or founding) population is of critical importance for successful invasion. Results are provided for one-dimensional and two-dimensional models. In the one-dimensional case, we consider initial conditions supported by two disjoint intervals of length L/2 and separated by a distance α. Analytical as well as numerical results indicate that the critical size L ∗(α) of the population, where the invasion is successful if and only if L>L ∗(α), is a continuous function of α and tends to increase with α, at least when α is not too small. This result emphasizes the detrimental effect of fragmentation. In the two-dimensional case, we consider more general, stochastically generated initial conditions u 0, and we provide a new and rigorous definition of the rate of fragmentation of u 0. We then conduct a statistical analysis of the probability of successful invasion depending on the size of the support of u 0 and the fragmentation rate of u 0. Our results show that the outcome of an invasion is almost completely determined by these two parameters. Moreover, we observe that the minimum abundance required for successful invasion tends to increase in a non-linear fashion with the fragmentation rate. This effect of fragmentation is enhanced as the strength of the Allee effect is increased.