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Distributions of different cell types derived from stochastic simulations. a Frequencies of cells having successful switching for each set of parameters ðk à 0 ; ψÞ. b Ratios of GMP cells to MEP cells when cells have successfully switched in a for each set of parameters ðk à 0 ; ψÞ. c Parameter sets of ðk à 0 ; ψÞ that generate stochastic simulations with four steady states as shown in Fig. 4 (yellow part) or with two or three states (blue part). d Violin plots of natural log normalised (expression level per cell +1) distributions for three genes in different cell states derived from stochastic simulations with parameters k à 0 ¼ 0:52 and ψ = 0.0005.
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Although multistability is an important dynamic property of a wide range of complex systems, it is still a challenge to develop mathematical models for realising high order multistability using realistic regulatory mechanisms. To address this issue, we propose a robust method to develop multistable mathematical models by embedding bistable models t...
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... show the boundary of parameter space, we also keep certain sets of parameter values with which simulations move to one specific stable state. Figure 5a gives proportions of simulations that have successful switching in 20,000 simulations. When the value of k à 0 is between 0.1 and 0.2, the displacement speed of GATA2 is low, which gives limited relief of negative regulation to PU.1, but GATA1 increases gradually due to GATA-switching and weak positive regulation from GATA2 to GATA1. ...
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... competition between GATA1 and PU.1 will lead cells to different lineages. When the value of k à 0 is relatively large but the value of ψ is relatively small, the increase of GATA1 is slow due to the smaller value of ψ in GATA-switching. However, the negative regulation from GATA2 to PU.1 declines rapidly due to the larger value of k à 0 . Thus, Fig. 5b shows that the combination of larger k à 0 and smaller ψ values allows more cells to move to the GMP lineage with high expression level of PU.1. If there is no winner in the competition between GATA1 and PU.1, the cell then moves to the state with low expression levels of three genes (namely LE3G). Figure 5c shows that, when the value ...
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... there is no winner in the competition between GATA1 and PU.1, the cell then moves to the state with low expression levels of three genes (namely LE3G). Figure 5c shows that, when the value of k à 0 is larger than 0.2, there are four types of simulations as shown in Fig. 5 for a set of k à 0 and ψ values. We use a MATLAB package 59 to give the violin plot for the expression distributions of three genes in three different cellular states. ...
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... more cells to move to the GMP lineage with high expression level of PU.1. If there is no winner in the competition between GATA1 and PU.1, the cell then moves to the state with low expression levels of three genes (namely LE3G). Figure 5c shows that, when the value of k à 0 is larger than 0.2, there are four types of simulations as shown in Fig. 5 for a set of k à 0 and ψ values. We use a MATLAB package 59 to give the violin plot for the expression distributions of three genes in three different cellular states. The violin plot is a combination of a box plot and a kernel density plot that illustrates data peaks. The violin plots in Fig. 5d match the experimental observations ...
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... there are four types of simulations as shown in Fig. 5 for a set of k à 0 and ψ values. We use a MATLAB package 59 to give the violin plot for the expression distributions of three genes in three different cellular states. The violin plot is a combination of a box plot and a kernel density plot that illustrates data peaks. The violin plots in Fig. 5d match the experimental observations very well 21 ...
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... of LE3G state. Supplementary Fig. 2 shows that, for a fixed value of parameter ψ, the frequency increases as the value of k à 0 increases. In addition, for a fixed value of k à 0 , the frequency decreases as the value of ψ increases. The variation of parameter ψ is much more important than that of parameter k à 0 . For the simulations showing in Fig. 5d, the frequency is 0.1080 with k à 0 ¼ 0:52 and ψ = 0.0005. Figure 5d and Supplementary Fig. 2 suggest that more cells remain in the LE3G or P1H (GMP) state if GATA2 leaves the chromatin site fast (i.e. a large k à 0 value) and the expression of GATA1 is slow (i.e. a small ψ value). However, if the expression of GATA1 is fast (i.e. a ...
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... the simulations showing in Fig. 5d, the frequency is 0.1080 with k à 0 ¼ 0:52 and ψ = 0.0005. Figure 5d and Supplementary Fig. 2 suggest that more cells remain in the LE3G or P1H (GMP) state if GATA2 leaves the chromatin site fast (i.e. a large k à 0 value) and the expression of GATA1 is slow (i.e. a small ψ value). However, if the expression of GATA1 is fast (i.e. a large ψ value), more cells will transit to G1H (MEP) state and the frequency of the LE3G state is low, which is consistent with the results in a recent study 60 . ...
Citations
... Bi-stable models were then embedded to achieve a tri-stable model, which was further modeled to encompass four mutually exclusive stable states. The findings from their modeling fitted experimental data [59]. We cannot, therefore, exclude that HSCs can process complex information regarding how they make a choice of lineage. ...
By the mid-1960s, hematopoietic stem cells (HSCs) were well described. They generate perhaps the most complex array of functionally mature cells in an adult organism. HSCs and their descendants have been studied extensively, and findings have provided principles that have been applied to the development of many cell systems. However, there are uncertainties about the process of HSC development. They center around when and how HSCs become affiliated with a single-cell lineage. A longstanding view is that this occurs late in development and stepwise via a series of committed oligopotent progenitor cells, which eventually give rise to unipotent progenitors. A very different view is that lineage affiliation can occur as early as within HSCs, and the development of these cells to a mature end cell is then a continuous process. A key consideration is the extent to which lineage-affiliated HSCs self-renew to make a major contribution to hematopoiesis. This review examines the above aspects in relation to our understanding of hematopoiesis.
... The success in finding parameters leading to multistability indicated that the proposed methodology is robust and adequate for complex GRNs. Also, it might present a scalable and straightforward alternative to previous proposals 74,75 . Despite our simplified model, we propose that further advances seeking to correlate the parameters with biological observation could help quantify malignant states. ...
We presented a method to find potential cancer attractors using single-cell RNA sequencing (scRNA-seq) data. We tested our method in a Glioblastoma Multiforme (GBM) dataset, an aggressive brain tumor presenting high heterogeneity. Using the cancer attractor concept, we argued that the GBM’s underlying dynamics could partially explain the observed heterogeneity, with the dataset covering a representative region around the attractor. Exploratory data analysis revealed promising GBM’s cellular clusters within a 3-dimensional marker space. We approximated the clusters’ centroid as stable states and each cluster covariance matrix as defining confidence regions. To investigate the presence of attractors inside the confidence regions, we constructed a GBM gene regulatory network, defined a model for the dynamics, and prepared a framework for parameter estimation. An exploration of hyperparameter space allowed us to sample time series intending to simulate myriad variations of the tumor microenvironment. We obtained different densities of stable states across gene expression space and parameters displaying multistability across different clusters. Although we used our methodological approach in studying GBM, we would like to highlight its generality to other types of cancer. Therefore, this report contributes to an advance in the simulation of cancer dynamics and opens avenues to investigate potential therapeutic targets.
... Note that bistability is a particular case of the multistability property of dynamic systems. According to [19], the implementation of the multistability property is a difficult problem and it is described by the regulatory blocks of mathematical models. Figures 5-6 illustrate the results of the analysis of the set of stationary points in the Marchuk-Petrov model. ...
This work is devoted to the technology developed by the authors that allows one for fixed values of parameters and tracing by parameters to calculate stationary solutions of systems with delay and analyze their stability. We discuss the results of applying this technology to the Marchuk–Petrov antiviral immune response model with parameter values corresponding to hepatitis B infection. The presence of bistability and hysteresis properties in this model is shown for the first time.
... The robustness of a system can be regarded as the potential of this system to remain in the current state. For realizing switches between different system states, a larger variation of model parameter(s) generally is needed if the model is more robust [Tian & Burrage, 2006;Wu et al., 2022]. ...
Tumor immune escape refers to the inability of the immune system to clear tumor cells, which is one of the major obstacles in designing effective treatment schemes for cancer diseases. Although clinical studies have led to promising treatment outcomes, it is imperative to design theoretical models to investigate the long-term treatment effects. In this paper, we develop a mathematical model to study the interactions among tumor cells, immune escape tumor cells, and T lymphocyte. The chimeric antigen receptor (CAR) T-cell therapy is also described by the mathematical model. Bifurcation analysis shows that there exists backward bifurcation and saddle-node bifurcation when the immune intensity is used as the bifurcation parameter. The proposed model also exhibits bistability when its parameters are located between the saddle-node threshold and backward bifurcation threshold. Sensitivity analysis is performed to illustrate the effects of different mechanisms on the backward bifurcation threshold and basic immune reproduction number. Simulation studies confirm the bifurcation analysis results and predict various types of treatment outcomes using different CAR T-cell therapy strengths. Analysis and simulation results show that the immune intensity can be used to control the tumor size, but it has no effect on the control of the immune escape tumor size. The introduction of the CAR T-cell therapy will reduce the immune escape tumor size and the treatment effect depends on the CAR T-cell therapy strength.
... It is the defining trait of a switch that allows the ability to achieve multiple states, without altering internal genetic content [1,2]. It has been observed in diverse biological contexts -lactose utilization in E. coli [3], flower morphogenesis in plants [4], multisite phosphorylation [5], haematopoiesis [6], and cancer cell plasticity [7,8]. Thus, decoding the emergent dynamics of underlying regulatory networks is crucial for mapping the cell-fate trajectories and for designing synthetic multistable circuits [9]. ...
... In case of the interaction being an activation, the hill function is further divided by the fold-change parameter corresponding to the respective interaction i.e., in case of an activation from node B to node A, the hill function would be, " ( , # , , l )/l . The default range of values for Hill coefficient in RACIPE is [1,6], but we chose the range of [6,10] for a TTr, because it allows for a bimodal distribution for the node expression levels, to segregate 'high' and 'low' states (Fig S1A). We used the default values of number of parameter sets (=10000) and number of initial conditions per parameter set (=1000) for our simulations, although similar behaviour was observed when taking a larger number of parameter sets and/or initial conditions (Fig S1B). ...
... In case of the interaction being an activation, the hill function is further divided by the fold-change parameter corresponding to the respective interaction i.e., in case of an activation from node B to node A, the hill function would be, " ( , # , , l )/l . The default range of values for Hill coefficient in RACIPE is [1,6], but we chose the range of [6,10] for a TTr, because it allows for a bimodal distribution for the node expression levels, to segregate 'high' and 'low' states (Fig S1A). We used the default values of number of parameter sets (=10000) and number of initial conditions per parameter set (=1000) for our simulations, although similar behaviour was observed when taking a larger number of parameter sets and/or initial conditions (Fig S1B). ...
Elucidating the emergent dynamics of complex regulatory networks enabling cellular differentiation is crucial to understand embryonic development and suggest strategies for synthetic circuit design. A well-studied network motif often driving cellular decisions is a toggle switch - a set of two mutually inhibitory lineage-specific transcription factors A and B. A toggle switch often enables two possible mutually exclusive states - (high A, low B) and (low A, high B) - from a common progenitor cell. However, the dynamics of networks enabling differentiation of more than two cell types from a progenitor cell is not well-studied. Here, we investigate the dynamics of four master regulators A, B, C and D inhibiting each other, thus forming a toggle tetrahedron. Our simulations show that a toggle tetrahedron predominantly allows for co-existence of six ‘double positive’ or hybrid states where two of the nodes are expressed relatively high as compared to the remaining two - (high A, high B, low C, low D), (high A, low B, high C, low D), (high A, low B, low C, high D), (low A, high B, high C, low D), (low A, low B, high C, high D) and (low A, high B, low C, high D). Stochastic simulations showed state-switching among these phenotypes, indicating phenotypic plasticity. Finally, we apply our results to understand the differentiation of naive CD4 ⁺ T cells into Th1, Th2, Th17 and Treg subsets, suggesting Th1/Th2/Th17/Treg decision-making to be a two-step process. Our results reveal multistable dynamics and establish the stable co-existence of hybrid cell-states, offering a potential explanation for simultaneous differentiation of multipotent naïve CD4+ T cells.
... It exists for gene regulatory networks [51,52], signaling pathways [53,54], and metabolic networks [55]. Multi-stability would allow HSCs to switch to an appropriate state to accord with the various changes to external influences, and modelling has revealed that it is important to HSCs choosing a cell lineage [56]. The TFs GATA1, GATA2, and PU-1 play essential roles in HSC and HPC development. ...
There is compelling evidence to support the view that the cell-of-origin for chronic myeloid leukemia is a hematopoietic stem cell. Unlike normal hematopoietic stem cells, the progeny of the leukemia stem cells are predominantly neutrophils during the disease chronic phase and there is a mild anemia. The hallmark oncogene for chronic myeloid leukemia is the BCR-ABLp210 fusion gene. Various studies have excluded a role for BCR-ABLp210 expression in maintaining the population of leukemia stem cells. Studies of BCR-ABLp210 expression in embryonal stem cells that were differentiated into hematopoietic stem cells and of the expression in transgenic mice have revealed that BCR-ABLp210 is able to veer hematopoietic stem and progenitor cells towards a myeloid fate. For the transgenic mice, global changes to the epigenetic landscape were observed. In chronic myeloid leukemia, the ability of the leukemia stem cells to choose from the many fates that are available to normal hematopoietic stem cells appears to be deregulated by BCR-ABLp210 and changes to the epigenome are also important. Even so, we still do not have a precise picture as to why neutrophils are abundantly produced in chronic myeloid leukemia.
... We add that a simpler system than the one we studied here, and that we published earlier as referred in the text ( [17]) has already proved to be instrumental in very recent bioengineering work ( [23]). ...
In this study, we focus on the non-local impact of saddles in a multiply connected gene regulation network. We find that so-called saddle-ghosts, that is to say the impact saddles impart on dynamics even if the saddles are remote, is significant and can be essentially dominating the nature of the dynamics the network presents. We focused our enquiry on an idealized five-gene auto-regulating and mutually repressive fully connected gene regulation network. This network is a compromise, for much higher gene-number networks, the analysis would be intractable while smaller gene-number networks would perhaps not exhibit the characteristics of “many being more than just the sum of the individuals” that sought-after nonlinear complex dynamics require. We use a combination of numerical simulations and theoretical analysis. We find that, in most of the interesting dynamical range of asymmetry of repression strength between gene-pairs, non-local saddles impact the dynamics by slowing the flow of heteroclinic cycles in multiple locations and the shape is affected. We study the slowdown behavior of these heteroclinic paths throughout the dynamical range of asymmetry. Their presence makes the system essentially exhibit multiple quasi-stable states, with rapid deterministic transitions between them. These findings may impact Biology as it pertains to the understanding of the evolution of gene regulation dynamics.
... Whether HSCs acquire a bias towards/affiliation to a developmental pathway by a process that is stochastic or deterministic has been debated for some time [80]. Recent mathematical modeling of how HSCs veer towards a lineage envisages a high order of multi-stability within HSCs [81], and HSCs gradually acquire uni-lineage priming [32,82,83] where noise and bursting gene expression play key roles. This and multi-stability fit with a continuum model to show how HSCs adopt a pathway, and are contradictory to a bi-stable, tree-like and dichotomous model. ...
The hematopoietic cell system is a complex ecosystem that meets the steady-state and emergency needs of the production of the mature blood cell types. Steady-state hematopoiesis replaces worn out cells, and the hematopoietic system is highly adaptive to needs during, for example, an infection or bleeding. Hematopoiesis is highly integrated and the cell hierarchy behaves in a highly social manner. The social tailoring of hematopoietic stem cells to needs includes the generation of cells that are biased towards a cell lineage; these cells remain versatile and can still adopt a different pathway having made a lineage “choice”, and some cytokines instruct the lineage fate of hematopoietic stem and progenitor cells. Leukemia stem cells, which may well often arise from the transformation of a hematopoietic stem cell, sustain the hierarchy of cells for leukemia. Unlike hematopoietic stem cells, the offspring of leukemia stem cells belongs to just one cell lineage. The human leukemias are classified by virtue of their differentiating or partially differentiating cells belonging to just one cell lineage. Some oncogenes set the fate of leukemia stem cells to a single lineage. Therefore, lineage restriction may be largely an attribute whereby leukemia stem cells escape from the normal cellular society. Additional antisocial behaviors are that leukemia cells destroy and alter bone marrow stromal niches, and they can create their own niches.
This work is devoted to the technology developed by the authors that allows one for fixed values of parameters and tracing by parameters to calculate stationary solutions of systems with delay and analyze their stability. We discuss the results of applying this technology to Marchuk-Petrov's antiviral immune response model with parameter values corresponding to hepatitis B infection. The presence of bistability and hysteresis properties in this model is shown for the first time.
