Modelling and analysis of time-lags in some basic patterns of cell proliferation

Self-adapting infectious dynamics on random networks

Self-adaptive dynamics occurs in many fields of research, such as socio-economics, neuroscience, or biophysics. We consider a self-adaptive modeling approach, where adaptation takes place within a set of strategies based on the history of the state of the system. This leads to piecewise deterministic Markovian dynamics coupled to a non-Markovian adaptive mechanism. We apply this framework to basic epidemic models (SIS, SIR) on random networks. We consider a co-evolutionary dynamical network where node-states change through the epidemics and network topology changes through the creation and deletion of edges. For a simple threshold base application of lockdown measures, we observe large regions in parameter space with oscillatory behavior, thereby exhibiting one of the most reduced mechanisms leading to oscillations. For the SIS epidemic model, we derive analytic expressions for the oscillation period from a pairwise closed model, which is validated with numerical simulations for random uniform networks. Furthermore, the basic reproduction number fluctuates around one indicating a connection to self-organized criticality.

Oscillation of Second-Order Differential Equations with Multiple and Mixed Delays under a Canonical Operator

In this work, we obtained new sufficient and necessary conditions for the oscillation of second-order differential equations with mixed and multiple delays under a canonical operator. Our methods could be applicable to find the sufficient and necessary conditions for any neutral differential equations. Furthermore, we proved the validity of the obtained results via particular examples. At the end of the paper, we provide the future scope of this study.

A novel mathematical model of heterogeneous cell proliferation

We present a novel mathematical model of heterogeneous cell proliferation where the total population consists of a subpopulation of slow-proliferating cells and a subpopulation of fast-proliferating cells. The model incorporates two cellular processes, asymmetric cell division and induced switching between proliferative states, which are important determinants for the heterogeneity of a cell population. As motivation for our model we provide experimental data that illustrate the induced-switching process. Our model consists of a system of two coupled delay differential equations with distributed time delays and the cell densities as functions of time. The distributed delays are bounded and allow for the choice of delay kernel. We analyse the model and prove the nonnegativity and boundedness of solutions, the existence and uniqueness of solutions, and the local stability characteristics of the equilibrium points. We find that the parameters for induced switching are bifurcation parameters and therefore determine the long-term behaviour of the model. Numerical simulations illustrate and support the theoretical findings, and demonstrate the primary importance of transient dynamics for understanding the evolution of many experimental cell populations.

Differential Roles of Environmental Enrichment in Alzheimer's Type of Neurodegeneration and Physiological Aging

Impairment of hippocampal adult neurogenesis in aging or degenerating brain is a well-known phenomenon caused by the shortage of brain stem cell pool, alterations in the local microenvironment within the neurogenic niches, or deregulation of stem cell development. Environmental enrichment (EE) has been proposed as a potent tool to restore brain functions, to prevent aging-associated neurodegeneration, and to cure neuronal deficits seen in neurodevelopmental and neurodegenerative disorders. Here, we report our data on the effects of environmental enrichment on hippocampal neurogenesis in vivo and neurosphere-forming capacity of hippocampal stem/progenitor cells in vitro. Two models - Alzheimer's type of neurodegeneration and physiological brain aging - were chosen for the comparative analysis of EE effects. We found that environmental enrichment greatly affects the expression of markers specific for stem cells, progenitor cells and differentiated neurons (Pax6, Ngn2, NeuroD1, NeuN) in the hippocampus of young adult rats or rats with Alzheimer's disease (AD) model but less efficiently in aged animals. Application of time-lag mathematical model for the analysis of impedance traces obtained in real-time monitoring of cell proliferation in vitro revealed that EE could restore neurosphere-forming capacity of hippocampal stem/progenitor cells more efficiently in young adult animals (fourfold greater in the control group comparing to the AD model group) but not in the aged rats (no positive effect of environmental enrichment at all). In accordance with the results obtained in vivo, EE was almost ineffective in the recovery of hippocampal neurogenic reserve in vitro in aged, but not in amyloid-treated or young adult, rats. Therefore, EE-based neuroprotective strategies effective in Aβ-affected brain could not be directly extrapolated to aged brain.

Design and fabrication of porous biodegradable scaffolds: a strategy for tissue engineering

Current strategies of tissue engineering are focused on the reconstruction and regeneration of damaged or deformed tissues by grafting of cells with scaffolds and biomolecules. Recently, much interest is given to scaffolds which are based on mimic the extracellular matrix that have induced the formation of new tissues. To return functionality of the organ, the presence of a scaffold is essential as a matrix for cell colonization, migration, growth, differentiation and extracellular matrix deposition, until the tissues are totally restored or regenerated. A wide variety of approaches has been developed either in scaffold materials and production procedures or cell sources and cultivation techniques to regenerate the tissues/organs in tissue engineering applications. This study has been conducted to present an overview of the different scaffold fabrication techniques such as solvent casting and particulate leaching, electrospinning, emulsion freeze-drying, thermally induced phase separation, melt molding and rapid prototyping with their properties, limitations, theoretical principles and their prospective in tailoring appropriate micro-nanostructures for tissue regeneration applications. This review also includes discussion on recent works done in the field of tissue engineering.

Delayed bet-hedging resilience strategies under environmental fluctuations

Many biological populations, such as bacterial colonies, have developed through evolution a protection mechanism, called bet hedging, to increase their probability of survival under stressful environmental fluctuation. In this context, the concept of preadaptation refers to a common type of bet-hedging protection strategy in which a relatively small number of individuals in a population stochastically switch their phenotypes to a dormant metabolic state in which they increase their probability of survival against potential environmental shocks. Hence, if an environmental shock took place at some point in time, preadapted organisms would be better adapted to survive and proliferate once the shock is over. In many biological populations, the mechanisms of preadaptation and proliferation present delays whose influence in the fitness of the population are not well understood. In this paper, we propose a rigorous mathematical framework to analyze the role of delays in both preadaptation and proliferation mechanisms in the survival of biological populations, with an emphasis on bacterial colonies. Our theoretical framework allows us to analytically quantify the average growth rate of a bet-hedging bacterial colony with stochastically delayed reactions with arbitrary precision. We verify the accuracy of the proposed method by numerical simulations and conclude that the growth rate of a bet-hedging population shows a nontrivial dependency on their preadaptation and proliferation delays. Contrary to the current belief, our results show that faster reactions do not, in general, increase the overall fitness of a biological population.

Modelling cell turnover in a complex tissue during development

The growth of organs results from proliferation within distinct cellular compartments. Organ development also involves transitions between cell types and variations in cell cycle duration as development progresses, and is regulated by a balance between entry into the compartment, proliferation of cells within the compartment, acquisition of quiescence and exit from that cell state via differentiation or death. While it is important to understand how environmental or genetic alterations can perturb such development, most approaches employed to date are descriptive rather than quantitative. This is because the identification and quantification of such parameters, while tractable in vitro, is challenging in the context of a complex tissue in vivo. Here we present a new framework for determining cell turnover in developing organs in vivo that combines cumulative cell-labelling and quantification of distinct cell-cycle phases without assuming homogeneity of behaviour within that compartment. A mathematical model is given that allows the calculation of cell cycle length in the context of a specific biological example and assesses the uncertainty of this calculation due to incomplete knowledge of cell cycle dynamics. This includes the development of a two population model to quantify possible heterogeneity of cell cycle length within a compartment and estimate the aggregate proliferation rate. These models are demonstrated on data collected from a progenitor cell compartment within the developing mouse kidney, the cap mesenchyme. This tissue was labelled by cumulative infusion, volumetrically quantified across time, and temporally analysed for the proportion of cells undergoing proliferation. By combining the cell cycle length predicted by the model with measurements of total cell population and mitotic rate, this approach facilitates the quantification of exit from this compartment without the need for a direct marker of that event. As a method specifically designed with assumptions appropriate to developing organs we believe this approach will be applicable to a range of developmental systems, facilitating estimations of cell cycle length and compartment behaviour that extend beyond simple comparisons of mitotic rates between normal and perturbed states.

Numerical Analysis of Explicit One-Step Methods for Stochastic Delay Differential Equations

We consider the problem of strong approximations of the solution of stochastic differential equations of Itô form with a constant lag in the argument. We indicate the nature of the equations of interest, and give a convergence proof in full detail for explicit one-step methods. We provide some illustrative numerical examples, using the Euler–Maruyama scheme.

Development of a three dimensional multiscale computational model of the human epidermis

Transforming Growth Factor (TGF-β1) is a member of the TGF-beta superfamily ligand-receptor network. and plays a crucial role in tissue regeneration. The extensive in vitro and in vivo experimental literature describing its actions nevertheless describe an apparent paradox in that during re-epithelialisation it acts as proliferation inhibitor for keratinocytes. The majority of biological models focus on certain aspects of TGF-β1 behaviour and no one model provides a comprehensive story of this regulatory factor's action. Accordingly our aim was to develop a computational model to act as a complementary approach to improve our understanding of TGF-β1. In our previous study, an agent-based model of keratinocyte colony formation in 2D culture was developed. In this study this model was extensively developed into a three dimensional multiscale model of the human epidermis which is comprised of three interacting and integrated layers: (1) an agent-based model which captures the biological rules governing the cells in the human epidermis at the cellular level and includes the rules for injury induced emergent behaviours, (2) a COmplex PAthway SImulator (COPASI) model which simulates the expression and signalling of TGF-β1 at the sub-cellular level and (3) a mechanical layer embodied by a numerical physical solver responsible for resolving the forces exerted between cells at the multi-cellular level. The integrated model was initially validated by using it to grow a piece of virtual epidermis in 3D and comparing the in virtuo simulations of keratinocyte behaviour and of TGF-β1 signalling with the extensive research literature describing this key regulatory protein. This research reinforces the idea that computational modelling can be an effective additional tool to aid our understanding of complex systems. In the accompanying paper the model is used to explore hypotheses of the functions of TGF-β1 at the cellular and subcellular level on different keratinocyte populations during epidermal wound healing.

Exploring hypotheses of the actions of TGF-beta1 in epidermal wound healing using a 3D computational multiscale model of the human epidermis

In vivo and in vitro studies give a paradoxical picture of the actions of the key regulatory factor TGF-beta1 in epidermal wound healing with it stimulating migration of keratinocytes but also inhibiting their proliferation. To try to reconcile these into an easily visualized 3D model of wound healing amenable for experimentation by cell biologists, a multiscale model of the formation of a 3D skin epithelium was established with TGF-beta1 literature-derived rule sets and equations embedded within it. At the cellular level, an agent-based bottom-up model that focuses on individual interacting units (keratinocytes) was used. This was based on literature-derived rules governing keratinocyte behavior and keratinocyte/ECM interactions. The selection of these rule sets is described in detail in this paper. The agent-based model was then linked with a subcellular model of TGF-beta1 production and its action on keratinocytes simulated with a complex pathway simulator. This multiscale model can be run at a cellular level only or at a combined cellular/subcellular level. It was then initially challenged (by wounding) to investigate the behavior of keratinocytes in wound healing at the cellular level. To investigate the possible actions of TGF-beta1, several hypotheses were then explored by deliberately manipulating some of these rule sets at subcellular levels. This exercise readily eliminated some hypotheses and identified a sequence of spatial-temporal actions of TGF-beta1 for normal successful wound healing in an easy-to-follow 3D model. We suggest this multiscale model offers a valuable, easy-to-visualize aid to our understanding of the actions of this key regulator in wound healing, and provides a model that can now be used to explore pathologies of wound healing.

A stochastic 1D nearest-neighbour automaton models early development of the brown alga Ectocarpus siliculosus

Early development of the filamentous brown alga Ectocarpus siliculosus (Dillwyn) Lyngbye involves two cell types that are arranged in a polymorphic, but constrained, pattern. The present study aimed to decipher the cellular processes responsible for the establishment of this pattern. Thorough observations characterised five different events of division and differentiation that occurred during the early development. The hypothesis that a local control is responsible for these processes was tested. To do so, Ectomat, a stochastic automaton in which each cell only interacts with its closest neighbour(s), was created. The probabilities for the five events were adjusted to fit to the observations. Simulations with Ectomat reconstructed most of the essential properties of the sporophyte development, in terms of cell-type proportion, relative position and growth dynamics. The whole organism properties emerged by applying local transition rules. In conclusion, no global position information system was required at this development stage. Randomly occurring cell events, driven by simple contact interactions, are sufficient to account for the early filament development and establishment of the cell-type pattern of E. siliculosus.

An integrated systems biology approach to understanding the rules of keratinocyte colony formation

Closely coupled in vitro and in virtuo models have been used to explore the self-organization of normal human keratinocytes (NHK). Although it can be observed experimentally, we lack the tools to explore many biological rules that govern NHK self-organization. An agent-based computational model was developed, based on rules derived from literature, which predicts the dynamic multicellular morphogenesis of NHK and of a keratinocyte cell line (HaCat cells) under varying extracellular Ca++ concentrations. The model enables in virtuo exploration of the relative importance of biological rules and was used to test hypotheses in virtuo which were subsequently examined in vitro. Results indicated that cell-cell and cell-substrate adhesions were critically important to NHK self-organization. In contrast, cell cycle length and the number of divisions that transit-amplifying cells could undergo proved non-critical to the final organization. Two further hypotheses, to explain the growth behaviour of HaCat cells, were explored in virtuo-an inability to differentiate and a differing sensitivity to extracellular calcium. In vitro experimentation provided some support for both hypotheses. For NHKs, the prediction was made that the position of stem cells would influence the pattern of cell migration post-wounding. This was then confirmed experimentally using a scratch wound model.

Mathematical Models for the Proliferation of Neural Stem/Progenitor Cells in Clonogenic Culture

Neurospheres are free-floating heterogeneous spheroid structures maintained in culture and have been used for in vitro expansion of neural stem/progenitor cells. But the growth characteristics and proliferation kinetics of cell population in neurospheres are poorly understood. Using clonogenic culture and immunocytochemistry, we observed the growth dynamics of cell populations in neurospheres and identified key parameters relating to the growth dynamics of cells in neurospheres, including the fraction of dividing cells, cell cycle duration, time delay of first division, and survival rate of cells. Based on these parameters, we established two mathematical models that describe kinetic features of neural stem/progenitor cells under clonogenic conditions. These models provide a powerful tool to explain and predict the experimental outcome and clarify the cell growth characteristics of neural stem/progenitor cells.

Activation sequence as a key factor in spatio-temporal optimization of myocardial function

Using one-dimensional models of myocardial tissue, implemented as chains of virtual ventricular muscle segments that are kinematically connected in series, we studied the role of the excitation sequence in spatio-temporal organization of cardiac function. Each model element was represented by a well-verified mathematical model of cardiac electro-mechanical activity. We found that homogeneous chains, consisting of identical elements, respond to non-simultaneous stimulation by generation of complex spatio-temporal heterogeneities in element deformation. These are accompanied by the establishment of marked gradients in local electro-mechanical properties of the elements (heterogeneity in action potential duration, Ca2+ transient characteristics and sarcoplasmic reticulum Ca2+ loading). In heterogeneous chains, composed of elements simulating fast and slow contracting cardiomyocytes from different transmural layers, we found that only activation sequences where stimulation of the slower elements preceded that of faster ones gave rise to optimization of the system's electro-mechanical function, which was confirmed experimentally. Based on the results obtained, we hypothesize that the sequence of activation of cardiomyocytes in different ventricular layers is one of the key factors of spatio-temporal organization of myocardium. Moreover, activation sequence and regional differences in intrinsic electro-mechanical properties of cardiac muscle must be matched in order to optimize myocardial function.

An empirical approach for delayed oscillator stability and parametric identification

This paper investigates a semi-empirical approach for determining the stability of systems that can be modelled by ordinary differential equations with a time delay. This type of model is relevant to biological oscillators, machining processes, feedback control systems and models for wave propagation and reflection, where the motion of the waves themselves is considered to be outside the system model. A primary aim is to investigate the extension of empirical Floquet theory to experimental or numerical data obtained from time-delayed oscillators. More specifically, the reconstructed time series from a numerical example and an experimental milling system are examined to obtain a finite number of characteristic multipliers from the reduced order dynamics. A secondary goal of this work is to demonstrate a benefit of empirical characteristic multiplier estimation by performing system identification on a delayed oscillator. The principal results from this study are the accurate estimation of delayed oscillator characteristic multipliers and the utilization the empirical results for parametric identification of model parameters. Combining these results with previous research on an experimental milling system provides a particularly relevant result—the first approach for identifying all model parameters for stability prediction directly from the cutting process vibration history.

Theoretical analysis of destabilization resonances in time-delayed stochastic second-order dynamical systems and some implications for human motor control

A linear stochastic delay differential equation of second order is studied that can be regarded as a Kramers model with time delay. An analytical expression for the stationary probability density is derived in terms of a Gaussian distribution. In particular, the variance as a function of the time delay is computed analytically for several parameter regimes. Strikingly, in the parameter regime close to the parameter regime in which the deterministic system exhibits Hopf bifurcations, we find that the variance as a function of the time delay exhibits a sequence of pronounced peaks. These peaks are interpreted as delay-induced destabilization resonances arising from oscillatory ghost instabilities. On the basis of the obtained theoretical findings, reinterpretations of previous human motor control studies and predictions for future human motor control studies are provided.

A comprehensive continuous-time model for the appearance of CGH signal due to chromosomal missegregations during mitosis

Aneuploidy, the gain or loss of large regions of the genome, is a common feature in cancer cells. Irregularities in chromosomal copy number caused by missegregations of chromosomes during mitosis can be visualized by cytogenetic techniques including fluorescence in situ hybridization (FISH), spectral karyotyping (SKY) and comparative genomic hybridization (CGH). In the current work, we consider the propagation of irregular copy numbers throughout a cell population as the individual cells progress through ordinary mitotic cell cycles. We use an algebraic model to track the different copy numbers as states in a stochastic process, based on the model of chromosome instability of Gusev, Kagansky, and Dooley, and consider the average copy number of a particular chromosome within a cell population as a function of the cell division rate. We review a number of mathematical models for determining the length of the cell cycle, including the Smith-Martin transition probability model and the 'sloppy size' model of Wheals, Tyson and Diekmann. The program MITOSIM simulates the growth of a population of cells using the aforementioned models of the cell cycle. MITOSIM allows the cell population to grow, with occasional resampling, until the average copy number of a given chromosome in the population reaches a preset threshold signifying a positive copy number alteration in this region. MITOSIM calculates the relationship between the missegregation rate and the growth rate of the cell population. This allows the user to test hypotheses regarding the effect chromosomal aberrations have upon the cell cycle, cell growth rates, and time to population dominance.

Use of stochastic models to assess the effect of environmental factors on microbial growth

We present a novel application of a stochastic ecological model to the study and analysis of microbial growth dynamics as influenced by environmental conditions in an extensive experimental data set. The model proved to be useful in bridging the gap between theoretical ideas in ecology and an applied problem in microbiology. The data consisted of recorded growth curves of Escherichia coli grown in triplicate in a base medium with all 32 possible combinations of five supplements: glucose, NH(4)Cl, HCl, EDTA, and NaCl. The potential complexity of 2(5) experimental treatments and their effects was reduced to 2(2) as just the metal chelator EDTA, the presumed osmotic pressure imposed by NaCl, and the interaction between these two factors were enough to explain the variability seen in the data. The statistical analysis showed that the positive and negative effects of the five chemical supplements and their combinations were directly translated into an increase or decrease in time required to attain stationary phase and the population size at which the stationary phase started. The stochastic ecological model proved to be useful, as it effectively explained and summarized the uncertainty seen in the recorded growth curves. Our findings have broad implications for both basic and applied research and illustrate how stochastic mathematical modeling coupled with rigorous statistical methods can be of great assistance in understanding basic processes in microbial ecology.

Experimental and mathematical study of the influence of growth factors on the growth kinetics of adult human articular chondrocytes

This study aimed at determining how kinetic parameters of adult human articular chondrocytes (AHAC) growth are modulated by the growth factor combination TGFbeta1, FGF-2, and PDGF BB (TFP), recently shown to stimulate AHAC proliferation. AHAC, isolated from cartilage biopsies of three individuals, were cultured in medium without (CTR) or with TFP. For growth curves, AHAC were seeded at 1,000 cells/cm(2) and cultured for 12 days, with cell numbers measured fluorimetrically in the same wells every 12 h. For microcolony tests, AHAC were seeded at 2.5 cells/cm(2) and cultured for 6 days, with cell numbers determined for each microcolony by phase contrast microscopy every 8 h. A mathematical model combining delay and logistic equations was developed to capture the growth kinetic parameters and to enable the description of the complete growth process of the cell culture. As compared to CTR medium, the presence of TFP increased the number of cells/well starting from the fifth day of culture, and a four-fold larger cell number was reached at confluency. For single microcolonies, TFP reduced the time for the first cell division by 26.6%, the time for subsequent cell divisions (generation time) by 16.8%, and the percentage of quiescent cells (Q(c)) by 42.5%. The mathematical model fitted well the experimental data of the growth kinetic. Finally, using both microcolony tests and the mathematical model, we determined that prolonged cell expansion induces an enrichment of AHAC with shorter first division time, but not of those with shorter generation time.

Discrete Event Modeling of CD4+ Memory T Cell Generation

Studies of memory T cell differentiation are hampered by a lack of quantitative models to test hypotheses in silico before in vivo experimentation. We created a stochastic computer model of CD4+ memory T cell generation that can simulate and track 10(1)-10(8) individual lymphocytes over time. Parameters for the model were derived from experimental data using naive human CD4+ T cells stimulated in vitro. Using discrete event computer simulation, we identified two key variables that heavily influence effector burst size and the persistent memory pool size: the cell cycle dependent probability of apoptosis, and the postactivation mitosis at which memory T cells emerge. Multiple simulations were performed and varying critical parameters permitted estimates of how sensitive the model was to changes in all of the model parameters. We then compared two hypotheses of CD4+ memory T cell generation: maturation from activated naive to effector to memory cells (model I) vs direct progression from activated naive to memory cells (model II). We find that direct progression of naive to memory T cells does not explain published measurements of the memory cell mass unless postactivation expansion of the memory cell cohort occurs. We conclude that current models suggesting direct progression of activated naive cells to the persistent memory phenotype (model II) do not account for the experimentally measured size of the postactivation CD4+, Ag-specific, memory T cell cohort.

Mathematical modelling of skeletal repair

Tissue engineering offers significant promise as a viable alternative to current clinical strategies for replacement of damaged tissue as a consequence of disease or trauma. Since mathematical modelling is a valuable tool in the analysis of complex systems, appropriate use of mathematical models has tremendous potential for advancing the understanding of the physical processes involved in such tissue reconstruction. In this review, the potential benefits, and limitations, of theoretical modelling in tissue engineering applications are examined with specific emphasis on tissue engineering of bone. A central tissue engineering approach is the in vivo implantation of a biomimetic scaffold seeded with an appropriate population of stem or progenitor cells. This review will therefore consider the theory behind a number of key factors affecting the success of such a strategy including: stem cell or progenitor population expansion and differentiation ex vivo; cell adhesion and migration, and the effective design of scaffolds; and delivery of nutrient to avascular structures. The focus will be on current work in this area, as well as on highlighting limitations and suggesting possible directions for future work to advance health-care for all.

Mechanisms of muscle stem cell expansion with cytokines

Stem cell expansion and proliferation are important for cell transplantation and stem cell-mediated applications. While we have demonstrated that muscle stem cells can be obtained from adult skeletal muscle tissue, these cells represent only a small percentage of the muscle-derived cells and require in vitro expansion for successful stem cell-mediated therapies. In this study, we have examined the potential of several cytokines to stimulate stem cell growth by combining a non-exponential mathematical model with a unique cell culture system. The growth kinetics of two populations of muscle stem cells were characterized in culture medium supplemented with epidermal growth factor (EGF), fibroblast growth factor-2 (FGF-2), insulin-like growth factor-1 (IGF-1), FLT-3 ligand, hepatocyte growth factor, or stem cell factor (SCF). The division time (DT) and fraction of mitotically active cells were investigated as key parameters to further understand the mechanism of the expansion of the stem cell populations. Our results show that expansion of the freshly isolated, muscle-derived stem cells (MDSC) occurred by recruiting cells into the cell cycle in the presence of EGF, IGF-1, and SCF. However, expansion of the cultured stem cell clone, MC13, is attributed to a reduction of the length of the cell cycle in the presence of FGF-2, EGF, IGF-1, and SCF. Both MDSC and MC13 growth were inhibited in the presence of FLT-3 ligand by increasing the length of the cell cycle. Our results suggest that EGF, IGF-1, FGF-2, and SCF are important cytokines for stimulating the proliferation of MDSC. In addition, this study illustrates that expansion of stem cells occurs through different mechanisms, which consequently demonstrates the importance of monitoring several parameters of cell growth, such as DT and dividing fraction, following stimulation with growth factors.