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Stem cell regeneration is a vital biological process in self-renewing tissues, governing development and tissue homeostasis. Gene regulatory network dynamics are pivotal in controlling stem cell regeneration and cell type transitions. However, integrating the quantitative dynamics of gene regulatory networks at the single-cell level with stem cell regeneration at the population level poses significant challenges. This study presents a computational framework connecting gene regulatory network dynamics with stem cell regeneration through a data-driven formulation of the inheritance function. The inheritance function captures epigenetic state transitions during cell division in heterogeneous stem cell populations. Our scheme allows the derivation of the inheritance function based on a hybrid model of cross-cell-cycle gene regulation network dynamics. The proposed scheme enables us to derive the inheritance function based on the hybrid model of cross-cell-cycle gene regulation network dynamics. By explicitly incorporating gene regulatory network structure, it replicates cross-cell-cycling gene regulation dynamics through individual-cell-based modeling. The numerical scheme holds the potential for extension to diverse gene regulatory networks, facilitating a deeper understanding of the connection between gene regulation dynamics and stem cell regeneration., Comment: 27 pages, 9 figures

The global public health landscape is perpetually challenged by the looming threat of infectious diseases. Central to addressing this concern is the imperative to prevent and manage disease transmission during pandemics, particularly in unique settings. This study addresses the transmission dynamics of infectious diseases within conference venues, presenting a computational model designed to simulate transmission processes within a condensed timeframe (one day), beginning with sporadic cases. Our model intricately captures the activities of individual attendees within the conference venue, encompassing meetings, rest intervals, and meal breaks. While meetings entail proximity seating, rest and lunch periods allow attendees to interact with diverse individuals. Moreover, the restroom environment poses an additional avenue for potential infection transmission. Employing an individual-based model, we meticulously replicated the transmission dynamics of infectious diseases, with a specific emphasis on close-contact interactions between infected and susceptible individuals. Through comprehensive analysis of model simulations, we elucidated the intricacies of disease transmission dynamics within conference settings and assessed the efficacy of control strategies to curb disease dissemination. Ultimately, our study proffers a numerical framework for assessing the risk of infectious disease transmission during short-duration conferences, furnishing conference organizers with valuable insights to inform the implementation of targeted prevention and control measures., Comment: 25 pages; 8 figures

Motivation: Single-cell RNA sequencing (scRNA-seq) is a groundbreaking technology extensively utilized in biological research, facilitating the examination of gene expression at the individual cell level within a given tissue sample. While numerous tools have been developed for scRNA-seq data analysis, the challenge persists in capturing the distinct features of such data and replicating virtual datasets that share analogous statistical properties. Results: Our study introduces a generative approach termed scRNA-seq Diffusion Transformer (scRDiT). This method generates virtual scRNA-seq data by leveraging a real dataset. The method is a neural network constructed based on Denoising Diffusion Probabilistic Models (DDPMs) and Diffusion Transformers (DiTs). This involves subjecting Gaussian noises to the real dataset through iterative noise-adding steps and ultimately restoring the noises to form scRNA-seq samples. This scheme allows us to learn data features from actual scRNA-seq samples during model training. Our experiments, conducted on two distinct scRNA-seq datasets, demonstrate superior performance. Additionally, the model sampling process is expedited by incorporating Denoising Diffusion Implicit Models (DDIM). scRDiT presents a unified methodology empowering users to train neural network models with their unique scRNA-seq datasets, enabling the generation of numerous high-quality scRNA-seq samples. Availability and implementation: https://github.com/DongShengze/scRDiT, Comment: 11 pages, 4 figures

In this paper we derive an upper bound for the degree of the strict invariant algebraic curve of a polynomial system in the complex project plane under generic condition. The results are obtained through the algebraic multiplicities of the system at the singular points. A method for computing the algebraic multiplicity using Newton polygon is also presented., Comment: 15pages, 1 figures

Stem cell regeneration is a crucial biological process for most self-renewing tissues during the development and maintenance of tissue homeostasis. In developing the mathematical models of stem cell regeneration and tissue development, cell division is the core process connecting different scale biological processes and leading to changes in cell population number and the epigenetic state of cells. This chapter focuses on the primary strategies for modeling cell division in biological systems. The Lagrange coordinate modeling approach considers gene network dynamics within each cell and random changes in cell states and model parameters during cell division. In contrast, the Euler coordinate modeling approach formulates the evolution of cell population numbers with the same epigenetic state via a differential-integral equation. These strategies focus on different scale dynamics, respectively, and result in two methods of modeling Waddington's epigenetic landscape: the Fokker-Planck equation and the differential-integral equation approaches. The differential-integral equation approach formulates the evolution of cell population density based on simple assumptions in cell proliferation, apoptosis, differentiation, and epigenetic state transitions during cell division. Moreover, machine learning methods can establish low-dimensional macroscopic measurements of a cell based on single-cell RNA sequencing data. The low dimensional measurements can quantify the epigenetic state of cells and become connections between static single-cell RNA sequencing data with dynamic equations for tissue development processes. The differential-integral equation presented in this chapter provides a reasonable approach to understanding the complex biological processes of tissue development and tumor progression., Comment: 46pages, 1 figures

Maintaining tissue homeostasis requires appropriate regulation of stem cell differentiation. The Waddington landscape posits that gene circuits in a cell form a potential landscape of different cell types, wherein cells follow attractors of the probability landscape to develop into distinct cell types. However, how adult stem cells achieve a delicate balance between self-renewal and differentiation remains unclear. We propose that random inheritance of epigenetic states plays a pivotal role in stem cell differentiation and present a hybrid model of stem cell differentiation induced by epigenetic modifications. Our comprehensive model integrates gene regulation networks, epigenetic state inheritance, and cell regeneration, encompassing multi-scale dynamics ranging from transcription regulation to cell population. Through model simulations, we demonstrate that random inheritance of epigenetic states during cell divisions can spontaneously induce cell differentiation, dedifferentiation, and transdifferentiation. Furthermore, we investigate the influences of interfering with epigenetic modifications and introducing additional transcription factors on the probabilities of dedifferentiation and transdifferentiation, revealing the underlying mechanism of cell reprogramming. This \textit{in silico} model provides valuable insights into the intricate mechanism governing stem cell differentiation and cell reprogramming and offers a promising path to enhance the field of regenerative medicine., Comment: 34 pages, 12 figures

This paper studies a random differential equation with random switch perturbations. We explore how the maximum displacement from the equilibrium state depends on the statistical properties of time series of {the} random switches. We show a power law dependence between the upper bound of displacement and the frequency of random perturbation switches, and the slope of power law dependence is dependent on the specific distribution of the intervals between switching times. This result {suggests} a quantitative connection between frequency modulation and amplitude modulation under random perturbations., Comment: 12 pages, 5 figures

Induced pluripotent stem cells (iPSCs) provide a great model to study the process of reprogramming and differentiation of stem cells. Single-cell RNA sequencing (scRNA-seq) enables us to investigate the reprogramming process at single-cell level. Here, we introduce single-cell entropy (scEntropy) as a macroscopic variable to quantify the cellular transcriptome from scRNA-seq data during reprogramming and differentiation of iPSCs. scEntropy measures the relative order parameter of genomic transcriptions at single cell level during the cell fate change process, which shows increasing during differentiation, and decreasing upon reprogramming. Moreover, based on the scEntropy dynamics, we construct a phenomenological stochastic differential equation model and the corresponding Fokker-Plank equation for cell state transitions during iPSC differentiation, which provide insights to infer cell fates changes and stem cell differentiation. This study is the first to introduce the novel concept of scEntropy to the biological process of iPSC, and suggests that the scEntropy can provide a suitable quantify to describe cell fate transition in differentiation and reprogramming of stem cells., Comment: 12 pages, 5 figures

We present the use of single-cell entropy (scEntropy) to measure the order of the cellular transcriptome profile from single-cell RNA-seq data, which leads to a method of unsupervised cell type classification through scEntropy followed by the Gaussian mixture model (scEGMM). scEntropy is straightforward in defining an intrinsic transcriptional state of a cell. scEGMM is a coherent method of cell type classification that includes no parameters and no clustering; however, it is comparable to existing machine learning-based methods in benchmarking studies and facilitates biological interpretation., Comment: 7 pages, 5 figures

During mammalian embryo development, reprogramming of DNA methylation plays important roles in the erasure of parental epigenetic memory and the establishment of na\"{i}ve pluripogent cells. Multiple enzymes that regulate the processes of methylation and demethylation work together to shape the pattern of genome-scale DNA methylation and guid the process of cell differentiation. Recent availability of methylome information from single-cell whole genome bisulfite sequencing (scBS-seq) provides an opportunity to study DNA methylation dynamics in the whole genome in individual cells, which reveal the heterogeneous methylation distributions of enhancers in embryo stem cells (ESCs). In this study, we developed a computational model of enhancer methylation inheritance to study the dynamics of genome-scale DNA methylation reprogramming during exit from pluripotency. The model enables us to track genome-scale DNA methylation reprogramming at single-cell level during the embryo development process, and reproduce the DNA methylation heterogeneity reported by scBS-seq. Model simulations show that DNA methylation heterogeneity is an intrinsic property driven by cell division along the development process, and the collaboration between neighboring enhancers is required for heterogeneous methylation. Our study suggest that the mechanism of genome-scale oscillation proposed by Rulands et al. (2018) might not necessary to the DNA methylation during exit from pluripotency., Comment: 25 pages, 4 figures

Predictive modeling of the evolutionary dynamics of cancer is a challenge issue in computational cancer biology. In this paper, we propose a general mathematical model framework for the evolutionary dynamics of cancer with plasticity and heterogeneity in cancer cells. Cancer is a group of diseases involving abnormal cell growth, during which abnormal regulations in stem cell regeneration are essential for the dynamics of cancer development. In general, the dynamics of stem cell regeneration can be simplified as a $\mathrm{G_0}$ phase cell cycle model, which lead to a delay differentiation equation. When cell heterogeneity and plasticity are considered, we establish a differential-integral equation based on the random transition of epigenetic states of stem cells during cell division. The proposed model highlights cell heterogeneity and plasticity, and connects the heterogeneity with cell-to-cell variance in cellular behaviors, e.g. proliferation, apoptosis, and differentiation/senescence, and can be extended to include gene mutation-induced tumor development. Hybrid computations models are developed based on the mathematical model framework, and are applied to the process of inflammation-induced tumorigenesis and tumor relapse after CAR-T therapy. Finally, we give rise to several mathematical problems related to the proposed differential-integral equation. Answers to these problems are crucial for the understanding of the evolutionary dynamics of cancer., Comment: 19 pages, 3 figures

Stem cell heterogeneity is essential for the homeostasis in tissue development. This paper established a general formulation for understanding the dynamics of stem cell regeneration with cell heterogeneity and random transitions of epigenetic states. The model generalizes the classical G0 cell cycle model, and incorporates the epigenetic states of stem cells that are represented by a continuous multidimensional variable and the kinetic rates of cell behaviors, including proliferation, differentiation, and apoptosis, that are dependent on their epigenetic states. Moreover, the random transition of epigenetic states is represented by an inheritance probability that can be described as a conditional beta distribution. This model can be extended to investigate gene mutation-induced tumor development. The proposed formula is a generalized formula that helps us to understand various dynamic processes of stem cell regeneration, including tissue development, degeneration, and abnormal growth., Comment: 36 pages, 7 figures

With the rapid development of urbanization, the boom of vehicle numbers has resulted in serious traffic accidents, which led to casualties and huge economic losses. The ability to predict the risk of traffic accident is important in the prevention of the occurrence of accidents and to reduce the damages caused by accidents in a proactive way. However, traffic accident risk prediction with high spatiotemporal resolution is difficult, mainly due to the complex traffic environment, human behavior, and lack of real-time traffic-related data. In this study, we collected big traffic accident data. By analyzing the spatial and temporal patterns of traffic accident frequency, we presented the spatiotemporal correlation of traffic accidents. Based on the patterns we found in analysis, we proposed a high accurate deep learning model based on recurrent neural network toward the prediction of traffic accident risk. The predictive accident risk can be potential applied to the traffic accident warning system. The proposed method can be integrated into an intelligent traffic control system toward a more reasonable traffic prediction and command organization.

Translation is a central biological process by which proteins are synthesized from genetic information contained within mRNAs. Here we study the kinetics of translation at molecular level through a stochastic simulation model. The model explicitly include RNA sequences, ribosome dynamics, tRNA pool and biochemical reactions in the translation elongation. The results show that the translation efficiency is mainly limited by the available ribosome number, translation initiation and the translation elongation time. The elongation time is log-normal distribution with mean and variance determined by both the codon saturation and the process of aa-tRNA selection at each codon binding. Moreover, our simulations show that the translation accuracy exponentially decreases with the sequence length. These results suggest that aa-tRNA competition is crucial for both translation elongation, translation efficiency and the accuracy, which in turn determined the effective protein production rate of correct proteins. Our results improve the dynamical equation of protein production with a delay differential equation which is dependent on sequence informations through both the effective production rate and the distribution of elongation time., Comment: 8 pages, 10 figures

The tumor suppressor p53 plays a central role in cell fate decisions after DNA damage. Programmed Cell Death 5 (PDCD5) is known to interact with the p53 pathway to promote cell apoptosis. Recombinant human PDCD5 can significantly sensitize different cancers to chemotherapies. In the present paper, we construct a computational model that includes PDCD5 interactions in the p53 signaling network and study the effects of PDCD5 on p53-mediated cell fate decisions during the DNA damage response. Our results revealed that PDCD5 functions as a co-activator of p53 that regulates p53-dependent cell fate decisions via the mediation of p53 dynamics. The effects of PDCD5 are dose-dependent such that p53 can display either sustained or pulsed dynamics at different PDCD5 levels. Moreover, PDCD5 regulates caspase-3 activation via two mechanisms during the two phases of sustained and pulsed p53 dynamics. This study provides insights regarding how PDCD5 functions as a regulator of the p53 pathway and might be helpful for increasing our understanding of the molecular mechanisms by which PDCD5 can be used to treat cancers., Comment: 12 pages, 7 figures

Dynamics of p53 is known to play important roles in the regulation of cell fate decisions in response to various stresses, and PDCD5 functions as a co-activator of p53 to modulate the p53 dynamics. In the present paper, we investigate how p53 dynamics are modulated by PDCD5 during the DNA damage response using methods of bifurcation analysis and potential landscape. Our results reveal that p53 activities can display rich dynamics under different PDCD5 levels, including monostability, bistability with two stable steady states, oscillations, and co-existence of a stable steady state and an oscillatory state. Physical properties of the p53 oscillations are further shown by the potential landscape, in which the potential force attracts the system state to the limit cycle attractor, and the curl flux force drives the coherent oscillation along the cyclic. We also investigate the effect of PDCD5 efficiency on inducing the p53 oscillations. We show that Hopf bifurcation is induced by increasing the PDCD5 efficiency, and the system dynamics show clear transition features in both barrier height and energy dissipation when the efficiency is close to the bifurcation point. This study provides a global picture of how PDCD5 regulates p53 dynamics via the interaction with the p53-Mdm2 oscillator and can be helpful in understanding the complicate p53 dynamics in a more complete p53 pathway., Comment: 11 pages, 8 figures

This paper studies the second moment boundedness of solutions of linear stochastic delay differential equations. First, we give a framework, for general $\mathrm{N}$-dimensional linear stochastic differential equations with a single discrete delay, of calculating the characteristic function for the second moment boundedness. Next, we apply the proposed framework to a special case of a type of 2-dimensional equation that the stochastic terms are decoupled. For the 2-dimensional equation, we obtain the characteristic function explicitly given by equation coefficients, the characteristic function gives sufficient conditions for the second moment to be bounded or unbounded., Comment: 39 pages

This paper studies the moment boundedness of solutions of linear stochastic delay differential equations with distributed delay. For a linear stochastic delay differential equation, the first moment stability is known to be identical to that of the corresponding deterministic delay differential equation. However, boundedness of the second moment is complicated and depends on the stochastic terms. In this paper, the characteristic function of the equation is obtained through techniques of Laplace transform. From the characteristic equation, sufficient conditions for the second moment to be bounded or unbounded are proposed., Comment: 38 pages, 2 figures

This paper considers adiabatic reduction in a model of stochastic gene expression with bursting transcription considered as a jump Markov process. In this model, the process of gene expression with auto-regulation is described by fast/slow dynamics. The production of mRNA is assumed to follow a compound Poisson process occurring at a rate depending on protein levels (the phenomena called bursting in molecular biology) and the production of protein is a linear function of mRNA numbers. When the dynamics of mRNA is assumed to be a fast process (due to faster mRNA degradation than that of protein) we prove that, with appropriate scalings in the burst rate, jump size or translational rate, the bursting phenomena can be transmitted to the slow variable. We show that, depending on the scaling, the reduced equation is either a stochastic differential equation with a jump Poisson process or a deterministic ordinary differential equation. These results are significant because adiabatic reduction techniques seem to have not been rigorously justified for a stochastic differential system containing a jump Markov process. We expect that the results can be generalized to adiabatic methods in more general stochastic hybrid systems., Comment: 17 pages

This paper studies the positive solutions of a class of delay differential equations with two delays. These equations originate from the modeling of hematopoietic cell populations. We give a sufficient condition on the initial function for $t\leq 0$ such that the solution is positive for all time $t>0$. The condition is "optimal". We also discuss the long time behavior of these positive solutions through a dynamical system on the space of continuous functions. We give a characteristic description of the $\omega$ limit set of this dynamical system, which can provide informations about the long time behavior of positive solutions of the delay differential equation., Comment: 15 pages, 2 figures

This paper addresses the question of how Brownian-like motion can arise from the solution of a deterministic differential delay equation. To study this we analytically study the bifurcation properties of an apparently simple differential delay equation and then numerically investigate the probabilistic properties of chaotic solutions of the same equation. Our results show that solutions of the deterministic equation with randomly selected initial conditions display a Gaussian-like density for long time, but the densities are supported on an interval of finite measure. Using these chaotic solutions as velocities, we are able to produce Brownian-like motions, which show statistical properties akin to those of a classical Brownian motion over both short and long time scales. Several conjectures are formulated for the probabilistic properties of the solution of the differential delay equation. Numerical studies suggest that these conjectures could be "universal" for similar types of "chaotic" dynamics, but we have been unable to prove this., Comment: 15 pages, 13 figures

Many cellular behaviors are regulated by gene regulation networks, kinetics of which is one of the main subjects in the study of systems biology. Because of the low number molecules in these reacting systems, stochastic effects are significant. In recent years, stochasticity in modeling the kinetics of gene regulation networks have been drawing the attention of many researchers. This paper is a self contained review trying to provide an overview of stochastic modeling. I will introduce the derivation of the main equations in modeling the biochemical systems with intrinsic noise (chemical master equation, Fokker-Plan equation, reaction rate equation, chemical Langevin equation), and will discuss the relations between these formulations. The mathematical formulations for systems with fluctuations in kinetic parameters are also discussed. Finally, I will introduce the exact stochastic simulation algorithm and the approximate explicit tau-leaping method for making numerical simulations., Comment: 25 pages, 4 figures

This paper gives a classification of first order polynomial differential operators of form $\mathscr{X} = X_1(x_1,x_2)\delta_1 + X_2(x_1,x_2)\delta_2$, $(\delta_i = \partial/\partial x_i)$. The classification is given through the order of an operator that is defined in this paper. Let $X=\mathscr{X}y$ to be the differential polynomial associated with $\mathscr{X}$, the order of $\mathscr{X}$, $\mathrm{ord}(\mathscr{X})$, is defined as the order of a differential ideal $\Lambda$ of differential polynomials that is a nontrivial expansion of the ideal $\{X\}$ and with the lowest order. In this paper, we prove that there are only four possible values for the order of a differential operator, 0, 1, 2, 3, or $\infty$. Furthermore, when the order is finite, the expansion $\Lambda$ is generated by $X$ and a differential polynomial $A$, which can be obtained through a rational solution of a partial differential equation that is given explicitly in this paper. When the order is infinite, the expansion $\Lambda$ is just the unit ideal. In additional, if, and only if, the order of $\mathscr{X}$ is 0, 1, or 2, the polynomial differential equation associating with $\mathscr{X}$ has Liouvillian first integrals. Examples for each class of differential operators are given at the end of this paper., Comment: 31 parges

In this paper, we introduce an approach to the protein folding problem from the point of view of statistical physics. Protein folding is a stochastic process by which a polypeptide folds into its characteristic and functional 3D structure from random coil. The process involves an intricate interplay between global geometry and local structure, and each protein seems to present special problems. We introduce CSAW (conditioned self-avoiding walk), a model of protein folding that combines the features of self-avoiding walk (SAW) and the Monte Carlo method. In this model, the unfolded protein chain is treated as a random coil described by SAW. Folding is induced by hydrophobic forces and other interactions, such as hydrogen bonding, which can be taken into account by imposing conditions on SAW. Conceptually, the mathematical basis is a generalized Langevin equation. To illustrate the flexibility and capabilities of the model, we consider several examples, including helix formation, elastic properties, and the transition in the folding of myoglobin. From the CSAW simulation and physical arguments, we find a universal elastic energy for proteins, which depends only on the radius of gyration $R_{g}$ and the residue number $N$. The elastic energy gives rise to scaling laws $R_{g}\sim N^{\nu}$ in different regions with exponents $\nu =3/5,3/7,2/5$, consistent with the observed unfolded stage, pre-globule, and molten globule, respectively. These results indicate that CSAW can serve as a theoretical laboratory to study universal principles in protein folding.

We propose a universal elastic energy for proteins, which depends only on the radius of gyration $R_{g}$ and the residue number $N$. It is constructed using physical arguments based on the hydrophobic effect and hydrogen bonding. Adjustable parameters are fitted to data from the computer simulation of the folding of a set of proteins using the CSAW (conditioned self-avoiding walk) model. The elastic energy gives rise to scaling relations of the form $R_{g}\sim N^{\nu}$ in different regions. It shows three folding stages characterized by the progression with exponents $\nu = 3/5, 3/7, 2/5$, which we identify as the unfolded stage, pre-globule, and molten globule, respectively. The pre-globule goes over to the molten globule via a break in behavior akin to a first-order phase transition, which is initiated by a sudden acceleration of hydrogen bonding.

In this paper we find a class of new degenerate central configurations and bifurcations in the Newtonian $n$-body problem. In particular we analyze the Rosette central configurations, namely a coplanar configuration where $n$ particles of mass $m_1$ lie at the vertices of a regular $n$-gon, $n$ particles of mass $m_2$ lie at the vertices of another $n$-gon concentric with the first, but rotated of an angle $\pi/n$, and an additional particle of mass $m_0$ lies at the center of mass of the system. This system admits two mass parameters $\mu=m_0/m_1$ and $\ep=m_2/m_1$. We show that, as $\mu$ varies, if $n> 3$, there is a degenerate central configuration and a bifurcation for every $\ep>0$, while if $n=3$ there is a bifurcations only for some values of $\epsilon$., Comment: 16 pages, 6 figures