Fig 1 - available via license: Creative Commons Attribution 4.0 International
Content may be subject to copyright.
Methodology for developing multistable models by embedding two sub-systems with bistability together. a Brief flowchart of hematopoietic hierarchy that is created with BioRender.com. HSCs hematopoietic stem cells, MPPs multipotent progenitors, MEPs megakaryocyte-erythroid progenitors, GMPs granulocyte-macrophage progenitors. b The principle of embeddedness: Z-U module is the first bistable sub-system. Once this module crosses the saddle point from state Z to state U, it enters the X-Y sub-system that has two stable steady states X and Y, reaching either state X or state Y via the auxiliary state U. c, d The structure of two double-negative feedback loops with positive autoregulations, which is the mechanisms for bistable sub-systems in HSCs. e The structure and mathematical model of regulatory network after embeddedness. The X-Y sub-system is embedded into the state U.
Source publication
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...
Contexts in source publication
Context 1
... motivation of this work is to develop a mathematical model to realise the tristable property of the HSC genetic regulatory network in Fig. 1a based on experimental observations. Figure 1b, e illustrates the embedding method to couple two bistable modules in a network together, where ' → ' and '⊣' denote the activating and inhibiting regulations, respectively. Variable U in the first Z-U module is an auxiliary node, which is assumed to be U = μX + δY, where μ and δ are two ...
Context 2
... models for GATA1-PU.1 and GATA-switching modules For the two double-negative feedback loops with positive autoregulation in Fig. 1c, d, we next develop two mathematical models for the Z-U module (13) and X-Y module (14). These two models have the same structure but with different model parameters. Theorem 1 shows that there are five possible nonnegative equilibria in these models. Theorem 2 indicates that two steady states located on the axis are stable under the ...
Context 3
... model parameters have the same values as the corresponding parameters in the Z-U module or the X-Y module. Supplementary Fig. 1 gives the 3D phase portrait of Fig. 2 Realisation of tristability by embedding two bistable sub-systems. a The phase plane of the toggle switch sub-system (1) with bistability (A and B: stable steady states, C: saddle state). ...
Context 4
... the assumption of a binary choice in each sub-module, the developed model is able to realise a rich variety of dynamics. Our research suggests that, depending on the properties of bistable systems, the embedding model of two bistable modules may have more than three stable steady states. In addition, using the embedding method in Fig. 1, the state U is not a meta-stable state but actually disappears from the system. Simulations show that, when the system leaves the high GATA2 expression state due to GATA-switching, genes GATA1 and PU.1 begin to increase their expression levels. Each stochastic simulation will reach one of the steady states with either high GATA1 ...
Context 5
... first develop a model for the network in Fig. 1c with bistability properties. Suppose that two sub-systems, namely the Z-U system and X-Y sub-system, have the same structure of a double-negative feedback loop and positive autoregulations. For the Z-U system, based on the formalism (8) with X = {z, u} ...
Context 6
... mathematical model for the network of three genes is formed by embedding the X-Y system into the Z-U system as shown in Fig. 1d. For simplicity, let u ¼ Hðx; yÞ = x + y. Since gene z is negatively regulated by gene u in sub-system (13), and u is a function of genes x and y, the expressions of genes x and y are also negatively regulated by gene z in the new embedding model. The non-linear vector fields G 1;2 ðx; y; Θ 1 ; tÞ are then transformed into new ...
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.
