One-hit stochastic decline in a mechanochemical model of cytoskeleton-induced neuron death II: transition state metastability

This is the second of two papers in which we study a mathematical model of cytoskeleton-induced neuron death. Recent evidence indicates that aggravated assembly or destruction of the cytoskeleton can trigger programmed death in neurons, by mechanisms as yet poorly understood. In our model, assembly control of the neuronal cytoskeleton interacts with both cellular stress levels and cytosolic free radical concentrations to trigger neurodegeneration. This trigger mechanism is further modulated by a diffusible toxic factor released from dying neurons. In the companion report we established that the model relates the observed general patterns of neuron decline to specific scales of cytoskeleton reorganization and cell-cell interaction strength. In this paper we study the transit of neurons through states intermediate between initial viability and cell death in our model. We find that the stochastic flow of neuron fate, from viability to cell death, self-organizes into two distinct temporal phases. There is a rapid relaxation of the initial neuron population to a more disordered phase that is long-lived, or metastable, with respect to the time scales of change in single cells. Strikingly, cellular egress from this metastable phase follows the one-hit kinetic pattern of exponential decline now established as a principal hallmark of cell death in neurodegenerative disorders. Intermediate state metastability may therefore be an important element in the systems biology of one-hit neurodegeneration.     

One-hit stochastic decline in a mechanochemical model of cytoskeleton-induced neuron death I: cell-fate arrival times

Much experimental evidence shows that the cytoskeleton is a downstream target and effector during cell death in numerous neurodegenerative diseases, including Parkinson's, Huntington's, and Alzheimer's diseases. However, recent evidence indicates that cytoskeletal dysfunction can also trigger neuronal death, by mechanisms as yet poorly understood. This is the first of two papers in which we study a mathematical model of cytoskeleton-induced neuron death. In our model, assembly control of the neuronal cytoskeleton interacts with both cellular stress levels and cytosolic free radical concentrations to trigger neurodegeneration. This trigger mechanism is further modulated by the presence of cell interactions in the form of a diffusible toxic factor released by dying neurons. We find that, consistent with empirical observations, our model produces one-hit exponential and sigmoid patterns of cell dropout. In all cases, cell dropout is exponential-tailed and described accurately by a gamma distribution. The transition between exponential and sigmoidal is gradual, and determined by a synergetic interaction between the magnitude of fluctuations in cytoskeleton assembly control and by the degree of cell coupling. We conclude that a single mechanism involving neuron interactions and fluctuations in cytoskeleton assembly control is compatible with the experimentally observed range of neuronal attrition kinetics.     

Biphasic growth in fish II: empirical assessment

In [Quince, et al., 2008. Biphasic growth in fish I: Theoretical foundations. J. Theor. Biol., doi:10.1016/j.jtbi.2008.05.029], we developed a set of biphasic somatic growth models, where maturation is accompanied by a deceleration of growth due to allocation of energy to reproduction. Here, we use growth data from both hatchery-raised and wild populations of a large freshwater fish (lake trout, Salvelinus namaycush) to test these models. We show that a generic biphasic model provides a better fit to these data than the von Bertalanffy model. We show that the observed deceleration of somatic growth in females varies directly with gonad weight at spawning, with observed egg volumes roughly 50% of the egg volumes predicted under the unrealistic assumption of perfectly efficient energy transfer from somatic lipids to egg lipids. We develop a Bayesian procedure to jointly fit a biphasic model to observed growth and maturity data. We show that two variants of the generic biphasic model, both of which assume that annual allocation to reproduction is adjusted to maximise lifetime reproductive output, provide complementary fits to wild population data: maturation time and early adult growth are best described by a model with no constraints on annual reproductive investment, while the growth of older fish is best described by a model that is constrained so that the ratio of gonad size to somatic weight (g) is fixed. This behaviour is consistent with the additional observation that g increases with size and age among younger, smaller breeding females but reaches a plateau among older, larger females. We then fit both of these optimal models to growth and maturation data from nineteen wild populations to generate population-specific estimates of ‘adapted mortality’ rate: the adult mortality consistent with observed growth and maturation schedules, given that both schedules are adapted to maximise lifetime reproductive output. We show that these estimates are strongly correlated with independent estimates of the adult mortality experienced by these populations.     

Fixation probabilities for the Moran process in evolutionary games with two strategies: graph shapes and large population asymptotics

This paper is based on the complete classification of evolutionary scenarios for the Moran process with two strategies given by Taylor et al. (Bull Math Biol 66(6):1621–1644, 2004. https://doi.org/10.1016/j.bulm.2004.03.004). Their classification is based on whether each strategy is a Nash equilibrium and whether the fixation probability for a single individual of each strategy is larger or smaller than its value for neutral evolution. We improve on this analysis by showing that each evolutionary scenario is characterized by a definite graph shape for the fixation probability function. A second class of results deals with the behavior of the fixation probability when the population size tends to infinity. We develop asymptotic formulae that approximate the fixation probability in this limit and conclude that some of the evolutionary scenarios cannot exist when the population size is large.

     

Biodiversity in model ecosystems, I: coexistence conditions for competing species

This is the first of two papers where we discuss the limits imposed by competition to the biodiversity of species communities. In this first paper, we study the coexistence of competing species at the fixed point of population dynamic equations. For many simple models, this imposes a limit on the width of the productivity distribution, which is more severe the more diverse the ecosystem is (1994, Theor. Popul. Biol. 45, 227-276). Here we review and generalize this analysis, beyond the "mean-field"-like approximation of the competition matrix used in previous works, and extend it to structured food webs. In all cases analysed, we obtain qualitatively similar relations between biodiversity and competition: the narrower the productivity distribution is, the more species can stably coexist. We discuss how this result, considered together with environmental fluctuations, limits the maximal biodiversity that a trophic level can host.     

On the basic reproduction number in a random environment

The concept of basic reproduction number $R_0$ in population dynamics is studied in the case of random environments. For simplicity the dependence between successive environments is supposed to follow a Markov chain. $R_0$ is the spectral radius of a next-generation operator. Its position with respect to 1 always determines population growth or decay in simulations, unlike another parameter suggested in a recent article (Hernandez-Suarez et al., Theor Popul Biol, doi:10.1016/j.tpb.2012.05.004, 2012). The position of the latter with respect to 1 determines growth or decay of the population’s expectation. $R_0$ is easily computed in the case of scalar population models without any structure. The main emphasis is on discrete-time models but continuous-time models are also considered.     

Genome-wide analysis of mammalian DNA segment fusion/fission

As a powerful tool for gene function prediction, gene fusion has been widely studied in prokaryotes and certain groups of eukaryotes, but it has been little applied in studies of mammalian genomes. With the first fully sequenced mammalian genomes (human, mouse, rat) now available, we defined and collected a set of fusion/fission event-linked segments (FFLS) based on structured organized genomic alignment. The statistics of the sequence features highlighted the FFLSs against their random context. We found that there are three groups of FFLSs with different component pairs (i.e. gene-gene, gene-noncoding and noncoding-noncoding) in all three mammalian genomes. The proteins encoded by the components of FFLSs in the first group shown a strong tendency to interact with each other. The segmental components in the last two groups which did not contain any protein-coding genes, were found not only to be transcribed to some level, but also more conserved than the random background. Thus, these segments are possibly carrying certain biologically functional elements. We propose that FFLS may be a potential tool for prediction and analysis of function and functional interaction of genetic elements, including both genes and noncoding elements, in mammalian genomes. The full list of the FFLSs in the genomes of the three mammals is available as supporting information at doi:10.1016/j.jtbi.2005.09.016.     

Simple signaling games of sexual selection (Grafen’s revisited)

We investigate several versions of a simple game of sexual selection, to explore the role of secondary sexual characters (the “handicap paradox”) with the tools of signaling theory. Our models admit closed form solutions. They are very much inspired by Grafen’s (J Theor Biol 144:517–546, 1990a; J Theor Biol 144:473–516, 1990b) seminal companion papers. By merging and simplifying his two approaches, we identify a not so minor artifact in the seminal study. We propose an alternative model to start with Grafen’s sexual selection theory, with several similarities with Getty (Anim Behav 56:127–130, 1998).     

Evolution of dispersal distance

The problem of how often to disperse in a randomly fluctuating environment has long been investigated, primarily using patch models with uniform dispersal. Here, we consider the problem of choice of seed size for plants in a stable environment when there is a trade off between survivability and dispersal range. Ezoe (J Theor Biol 190:287–293, 1998) and Levin and Muller-Landau (Evol Ecol Res 2:409–435, 2000) approached this problem using models that were essentially deterministic, and used calculus to find optimal dispersal parameters. Here we follow Hiebeler (Theor Pop Biol 66:205–218, 2004) and use a stochastic spatial model to study the competition of different dispersal strategies. Most work on such systems is done by simulation or nonrigorous methods such as pair approximation. Here, we use machinery developed by Cox et al. (Voter model perturbations and reaction diffusion equations 2011) to rigorously and explicitly compute evolutionarily stable strategies.     

Daphnias: from the individual based model to the large population equation

The class of deterministic ‘Daphnia’ models treated by Diekmann et al. (J Math Biol 61:277–318, 2010) has a long history going back to Nisbet and Gurney (Theor Pop Biol 23:114–135, 1983) and Diekmann et al. (Nieuw Archief voor Wiskunde 4:82–109, 1984). In this note, we formulate the individual based models (IBM) supposedly underlying those deterministic models. The models treat the interaction between a general size-structured consumer population (‘Daphnia’) and an unstructured resource (‘algae’). The discrete, size and age-structured Daphnia population changes through births and deaths of its individuals and through their aging and growth. The birth and death rates depend on the sizes of the individuals and on the concentration of the algae. The latter is supposed to be a continuous variable with a deterministic dynamics that depends on the Daphnia population. In this model setting we prove that when the Daphnia population is large, the stochastic differential equation describing the IBM can be approximated by the delay equation featured in (Diekmann et al., loc. cit.).     

γ-turn types prediction in proteins using the two-stage hybrid neural discriminant model

Due to the slightly success of protein secondary structure prediction using the various algorithmic and non-algorithmic techniques, similar techniques have been developed for predicting γ-turns in proteins by Kaur and Raghava [2003. A neural-network based method for prediction of γ-turns in proteins from multiple sequence alignment. Protein Sci. 12, 923-929]. However, the major limitation of previous methods was inability in predicting γ-turn types. In a recent investigation we introduced a sequence based predictor model for predicting γ-turn types in proteins [Jahandideh, S., Sabet Sarvestani, A., Abdolmaleki, P., Jahandideh, M., Barfeie, M, 2007a. γ-turn types prediction in proteins using the support vector machines. J. Theor. Biol. 249, 785-790]. In the present work, in order to analyze the effect of sequence and structure in the formation of γ-turn types and predicting γ-turn types in proteins, we applied novel hybrid neural discriminant modeling procedure. As the result, this study clarified the efficiency of using the statistical model preprocessors in determining the effective parameters. Moreover, the optimal structure of neural network can be simplified by a preprocessor in the first stage of hybrid approach, thereby reducing the needed time for neural network training procedure in the second stage and the probability of overfitting occurrence decreased and a high precision and reliability obtained in this way.     

Suppression of programmed cell death regulates the cyclical degeneration of organs in a colonial urochordate

The survival of animal tissues and organs is controlled through both activation and suppression of programmed cell death. In the colonial urochordate Botryllus schlosseri, the entire parental generation of zooids in a colony synchronously dies every week as the asexually derived generation of buds reaches functional maturity. This process, called takeover, involves massive programmed cell death (PCD) of zooid organs via apoptosis followed by programmed removal of cell corpses by blood phagocytes within approximately 1 day. We have previously reported that developing buds in conjunction with circulating phagocytes are key effectors of zooid resorption and macromolecular recycling during takeover, and as such engineer the reconstitution of a functional asexual generation every week [Lauzon, R.J., Ishizuka, K.J., Weissman, I.L., 2002. Cyclical generation and degeneration of organs in a colonial urochordate involves crosstalk between old and new: a model for development and regeneration. Dev. Biol. 249, 333-348]. Here, we demonstrate that zooid lifespan during cyclic blastogenesis is regulated by two independent signals: a bud-independent signal that activates zooid PCD and a bud-dependent, survival signal that acts in short-range fashion via the colonial vasculature. As zooids represent a transient, mass-produced commodity during Botryllus asexual development, PCD regulation in this animal via both activation and suppression enables it to remove and recycle its constituent zooids earlier when intra-colony resources are low, while maintaining the functional filter-feeding state when resources are adequate. We propose that this crosstalk mechanism between bud and parent optimizes survival of a B. schlosseri colony with each round of cyclic blastogenesis.     

Distinct destructive signal pathways of neuronal death in Alzheimer's disease

Abundant neuron loss is a major feature of Alzheimer's disease (AD). Hypotheses for this loss include abnormal amyloid precursor protein processing (i.e. excess Abeta production, protein aggregation or misfolding), oxidative stress, excitotoxicity and inflammation. Neuron loss is a major cause of dementia in AD; however, it seems that there is no definitive pathway that causes cell death in the AD brain. Here, we examine the hypotheses for neuron loss in AD and pose the argument that the means by which neurons degenerate is irrelevant for cognitive decline. The best treatment for cognitive decline is to prevent the toxicity that first sets the neuron on its path to destruction, which is the production of Abeta peptide.     

Duality,ancestral and diffusion processes in models with selection

The ancestral selection graph in population genetics was introduced by Krone and Neuhauser [Krone, S.M., Neuhauser, C., 1997. Ancestral process with selection. Theor. Popul. Biol. 51, 210–237] as an analogue of the coalescent genealogy of a sample of genes from a neutrally evolving population. The number of particles in this graph, followed backwards in time, is a birth and death process with quadratic death and linear birth rates. In this paper an explicit form of the probability distribution of the number of particles is obtained by using the density of the allele frequency in the corresponding diffusion model obtained by Kimura [Kimura, M., 1955. Stochastic process and distribution of gene frequencies under natural selection. Cold Spring Harbor Symposia on Quantitative Biology 20, 33–53]. It is shown that the process of fixation of the allele in the diffusion model corresponds to convergence of the ancestral process to its stationary measure. The time to fixation of the allele conditional on fixation is studied in terms of the ancestral process.     

Stochastic Modeling of Gene Regulation by Noncoding Small RNAs in the Strong Interaction Limit

Noncoding small RNAs (sRNAs) are known to play a key role in regulating diverse cellular processes, and their dysregulation is linked to various diseases such as cancer. Such diseases are also marked by phenotypic heterogeneity, which is often driven by the intrinsic stochasticity of gene expression. Correspondingly, there is significant interest in developing quantitative models focusing on the interplay between stochastic gene expression and regulation by sRNAs. We consider the canonical model of regulation of stochastic gene expression by sRNAs, wherein interaction between constitutively expressed sRNAs and mRNAs leads to stoichiometric mutual degradation. The exact solution of this model is analytically intractable given the nonlinear interaction term between sRNAs and mRNAs, and theoretical approaches typically invoke the mean-field approximation. However, mean-field results are inaccurate in the limit of strong interactions and low abundances; thus, alternative theoretical approaches are needed. In this work, we obtain analytical results for the canonical model of regulation of stochastic gene expression by sRNAs in the strong interaction limit. We derive analytical results for the steady-state generating function of the joint distribution of mRNAs and sRNAs in the limit of strong interactions and use the results derived to obtain analytical expressions characterizing the corresponding protein steady-state distribution. The results obtained can serve as building blocks for the analysis of genetic circuits involving sRNAs and provide new insights into the role of sRNAs in regulating stochastic gene expression in the limit of strong interactions.     

Mathematical modeling of viral kinetics under immune control during primary HIV-1 infection

Primary human immunodeficiency virus (HIV) infection is characterized by an initial exponential increase of viral load in peripheral blood reaching a peak, followed by a rapid decline to the viral setpoint. Although the target-cell-limited model can account for part of the viral kinetics observed early in infection [Phillips, 1996. Reduction of HIV concentration during acute infection: independence from a specific immune response. Science 271 (5248), 497-499], it frequently predicts highly oscillatory kinetics after peak viremia, which is not typically observed in clinical data. Furthermore, the target-cell-limited model is unable to predict long-term viral kinetics, unless a delayed immune effect is assumed [Stafford et al., 2000. Modeling plasma virus concentration during primary HIV infection. J. Theor. Biol. 203 (3), 285-301]. We show here that extending the target-cell-limited model, by implementing a saturation term for HIV-infected cell loss dependent upon infected cell levels, is able to reproduce the diverse observed viral kinetic patterns without the assumption of a delayed immune response. Our results suggest that the immune response may have significant effect on the control of the virus during primary infection and may support experimental observations that an anti-HIV immune response is already functional during peak viremia.     

Estimation of Cell Proliferation Dynamics Using CFSE Data

Advances in fluorescent labeling of cells as measured by flow cytometry have allowed for quantitative studies of proliferating populations of cells. The investigations (Luzyanina et al. in J. Math. Biol. 54:57–89, 2007; J. Math. Biol., 2009; Theor. Biol. Med. Model. 4:1–26, 2007) contain a mathematical model with fluorescence intensity as a structure variable to describe the evolution in time of proliferating cells labeled by carboxyfluorescein succinimidyl ester (CFSE). Here, this model and several extensions/modifications are discussed. Suggestions for improvements are presented and analyzed with respect to statistical significance for better agreement between model solutions and experimental data. These investigations suggest that the new decay/label loss and time dependent effective proliferation and death rates do indeed provide improved fits of the model to data. Statistical models for the observed variability/noise in the data are discussed with implications for uncertainty quantification. The resulting new cell dynamics model should prove useful in proliferation assay tracking and modeling, with numerous applications in the biomedical sciences.     

Mathematical modelling of juxtacrine patterning

Spatial pattern formation is one of the key issues in developmental biology. Some patterns arising in early development have a very small spatial scale and a natural explanation is that they arise by direct cell—cell signalling in epithelia. This necessitates the use of a spatially discrete model, in contrast to the continuum-based approach of the widely studied Turing and mechanochemical models. In this work, we consider the pattern-forming potential of a model for juxtacrine communication, in which signalling molecules anchored in the cell membrane bind to and activate receptors on the surface of immediately neighbouring cells. The key assumption is that ligand and receptor production are both up-regulated by binding. By linear analysis, we show that conditions for pattern formation are dependent on the feedback functions of the model. We investigate the form of the pattern: specifically, we look at how the range of unstable wavenumbers varies with the parameter regime and find an estimate for the wavenumber associated with the fastest growing mode. A previous juxtacrine model for Delta-Notch signalling studied by Collier et al. (1996, J. Theor. Biol. 183, 429–446) only gives rise to patterning with a length scale of one or two cells, consistent with the fine-grained patterns seen in a number of developmental processes. However, there is evidence of longer range patterns in early development of the fruit fly Drosophila. The analysis we carry out predicts that patterns longer than one or two cell lengths are possible with our positive feedback mechanism, and numerical simulations confirm this. Our work shows that juxtacrine signalling provides a novel and robust mechanism for the generation of spatial patterns.     

Performance evaluation of a novel conceptual bioprocess for clinically-required mass production of hematopoietic cells

Objective

The novel engineered bioprocess, which was designed and modeled to provide the clinically relevant cell numbers for different therapies in our previous work (Kaleybar et al. Food Bioprod Process 122:254–268, https://doi.org/10.1016/j.fbp.2020.04.012, 2020), was evaluated by using U937 as hematopoietic model cells.

Results

The culture system showed a 30-fold expansion of U937 cells in one-step during a 10-day culture period. The cell growth profile, the substrate and oxygen consumptions, and byproduct formations were all in agreement with the model predications during 7 days. The cell proliferation decrease after 7 days was attributed to optional oxygen limiting condition in the last days of culture. The bioreactor culture system revealed also a slight enhancement of lactate dehydrogenase (LDH) production as compared to the 2D conventional culture system, indicating the low impact of shear stress on cellular damage in the dynamic system.

Conclusions

The results demonstrated that the conceptual bioprocess for suspended stem cell production has a great potential in practice although additional experiments are required to improve the system.

     

Multiscale analysis of pattern formation via intercellular signalling

Lateral inhibition, a juxtacrine signalling mechanism by which a cell adopting a particular fate inhibits neighbouring cells from doing likewise, has been shown to be a robust mechanism for the formation of fine-grained spatial patterns (in which adjacent cells in developing tissues diverge to achieve contrasting states of differentiation), provided that there is sufficiently strong feedback. The fine-grained nature of these patterns poses problems for analysis via traditional continuum methods since these require that significant variation takes place only over lengthscales much larger than an individual cell and such systems have therefore been investigated primarily using discrete methods. Here, however, we apply a multiscale method to derive systematically a continuum model from the discrete Delta-Notch signalling model of Collier et al. (J.R. Collier, N.A.M. Monk, P.K. Maini, J.H. Lewis, Pattern formation by lateral inhibition with feedback: a mathematical model of Delta-Notch intercellular signalling, J. Theor. Biol., 183, 1996, 429-446) under particular assumptions on the parameters, which we use to analyse the generation of fine-grained patterns. We show that, on the macroscale, the contact-dependent juxtacrine signalling interaction manifests itself as linear diffusion, motivating the use of reaction-diffusion-based models for such cell-signalling systems. We also analyse the travelling-wave behaviour of our system, obtaining good quantitative agreement with the discrete system.