A robust method for designing multistable systems by embedding bistable subsystems

Siyuan Wu; Tianshou Zhou; Tianhai Tian

doi:10.1038/s41540-022-00220-1

A robust method for designing multistable systems by embedding bistable subsystems

Wu, Siyuan; Zhou, Tianshou; Tian, Tianhai 2022-03-25 00:00:00 www.nature.com/npjsba ARTICLE OPEN A robust method for designing multistable systems by embedding bistable subsystems 1 2 1✉ Siyuan Wu , Tianshou Zhou and Tianhai Tian 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 together. Using the GATA1-GATA2- PU.1 module in hematopoiesis as the test system, we ﬁrst develop a tristable model based on two bistable models without any high cooperative coefﬁcients, and then modify the tristable model based on experimentally determined mechanisms. The modiﬁed model successfully realises four stable steady states and accurately reﬂects a recent experimental observation showing four transcriptional states. In addition, we develop a stochastic model, and stochastic simulations successfully realise the experimental observations in single cells. These results suggest that the proposed method is a general approach to develop mathematical models for realising multistability and heterogeneity in complex systems. npj Systems Biology and Applications (2022) 8:10 ; https://doi.org/10.1038/s41540-022-00220-1 INTRODUCTION suggested that GATA2 and GATA1 sequentially bind the same 29,30 cis-elements, which is referred to as the GATA-switching . Multistability is the characteristic of a system that exhibits two or Mathematical modelling is a powerful tool to accurately more mutually exclusive stable states. This phenomenon has been describe the dynamics of hematopoiesis and to explore the observed in many different disciplines of science, including 1–4 5–8 regulatory mechanisms for controlling the transitions between genetic regulatory networks , cell signalling pathways , meta- 31–37 9 10,11 12 different cell types . For the GATA1-PU.1 module, Hill equations bolic networks , ecosystems , neuroscience , laser sys- 13,14 15 with high cooperativity were initially used to realise tristability .In tems , and quantum systems . When external and/or addition, mathematical models have been proposed to achieve internal conditions change, the system may switch from one bistability in gene regulatory networks without any high steady state to another either randomly by perturbations or in a 39,40 cooperativity coefﬁcients . Bifurcation theory is also an efﬁcient desired way according to the control strategies. In recent years method to explore the mechanisms of GATA1-PU.1 module .We mathematical models with multistability have been developed for have proposed a mathematical model to realise the mechanisms theoretical analysis and computer simulations, which shed light on of GATA-switching and designed an effective algorithm to realise the mechanisms that generate multistability and control the 16–19 tristability of mathematical models . Moreover, the underlying transition between steady states . mechanisms of how the stem/progenitor cells leave the stable As one of the important molecular systems showing multi- steady states and commit to a speciﬁc lineage were also revealed stability, hematopoiesis is a highly integrated developmental with the assistance of mathematical models . At the single cell process that controls the proliferation, differentiation and 20,21 level, the differentiation processes of embryonic stem cells were maturation of hematopoietic stem cells (HSCs) . HSCs have simulated by Langevin equations, which helped to identify the features of self-renewal and multipotency as well as the ability to differentiate into multipotent progenitors (MPPs). Each of these potential transcriptional regulators of lineage decision and commitment . Mathematical models have also been used to cell types is regarded as a stable state of the multistable system. In study the dynamical properties of diseases such as periodic addition, the formation of white and red blood cells is a dynamical haematological disorders . process that transits a cell from one stable cell type to another. Although these attempts have realised tristability by using This process begins with the differentiation of HSCs and enters the different assumptions, it is still a challenge to develop mathema- main stage at which cells reach either common myeloid 22,23 tical models to realise tristability using both the realistic regulatory progenitors (CMPs) or common lymphoid progenitors (CLPs) . Transcription factors play a key role in controlling the process of mechanisms and experimental data. On the other hand, blood cell lineage speciﬁcation. Experimental studies have substantial research studies have been conducted to develop 3,46–51 demonstrated that the genetic module GATA1-PU.1 is a vital mathematical models for realising bistability properties . Thus, the question is whether we can develop mathematical component for the fate commitment of CMPs between erythro- 24,25 models with tristability or higher order of multistability by using poiesis and granulopoiesis . HSCs are more likely to choose megakaryocyte/erythroid progenitors (MEPs) with high expression bistable models. To address this issue, we propose a robust levels of GATA1 , or conversely to choose granulocyte/macro- method to develop multistable models by embedding bistable phage progenitors (GMPs) with high expression levels of PU.1 .In models together. Using the GATA1-GATA2-PU.1 module as a addition, the regulation between genes GATA1 and GATA2 is an testing model, we develop a tristable model based on two essential driver of hematopoiesis . Experimental studies systems that have no high cooperativity coefﬁcients. 1 2 School of Mathematics, Monash University, Melbourne, VIC, Australia. School of Mathematics and Statistics, Sun Yet-Sen University, Guangzhou, China. email: Tianhai.Tian@monash.edu Published in partnership with the Systems Biology Institute 1234567890():,; S. Wu et al. Fig. 1 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. RESULTS It is assumed that the ﬁrst Z-U module follows model (1) and the second X-Y module satisﬁes the same model with same parameter Embedding method for designing multistable models space Θ , but different variables x and y, given by The motivation of this work is to develop a mathematical model to dx 4 realise the tristable property of the HSC genetic regulatory network ¼G ðx; y; Θ ; tÞ¼ 0:2 þ x; 1 1 3 dt 1þy (2) in Fig. 1a based on experimental observations. Figure 1b, e dy ¼G ðx; y; Θ ; tÞ¼ 0:2 þ y: 2 1 3 dt 1þx illustrates the embedding method to couple two bistable modules in a network together, where ’→ ’ and ’⊣’ denote the activating Now we embed these two sub-systems together using and inhibiting regulations, respectively. Variable U in the ﬁrst Z-U u ¼Hðx; yÞ¼ x þ y. Since gene z is negatively regulated by gene module is an auxiliary node, which is assumed to be U= μX + δY, u in the sub-system (1), and u is a function of genes x and y, the where μ and δ are two positive parameters. When the system stays expressions of genes x and y are also negatively regulated by gene in the state with a high expression level of Z and a low level of U, z in the new embedding model. Then the non-linear vector ﬁelds the expression levels of X and Y are low. However, when the G ðx; y; Θ ; tÞ are transformed into new non-linear vector ﬁelds 1;2 1 R ðx; y; z; Θ ; tÞ, respectively, which include genes x, y and z system has a low expression level of Z and a high level of U, the 1;2 1 from two sub-systems with negative regulations from gene z to system triggers the second module X-Y to choose either a high genes x and y. Therefore, the new model with three variables is level of X and a low level of Y or a low level of X and a high level of given by Y. In this way we realise the system with three stable states in dx 4 which one of the three variables (namely Z, X or Y) is at the high ¼R ðx; y; z; Θ ; tÞ¼ 0:2 þ x; 1 1 3 3 dt ð1þy Þð1þz Þ expression state but the other two are at low expression states. dy ¼R ðx; y; z; Θ ; tÞ¼ 0:2 þ y; w 1 3 3 (3) To demonstrate the effectiveness of the proposed embedding dt ð1þx Þð1þz Þ method, we use the toggle switch network as the test system . dz 4 ¼F ðz; u ¼ x þ y; Θ ; tÞ¼ 0:2 þ z: 1 1 3 dt 1þðxþyÞ This network consists of two genes that form a double negative feedback loop and is modelled by the following equations with Figure 2a shows the phase plane of the toggle switch sub-system parameter space Θ = {a = 0.2, b = 4, c = 3}, given by (1) with bistability properties, and Fig. 2b provides the 3D phase portrait of the embedded model (3) with three stable steady dz 4 states. The embedded model successfully realised the tristability, ¼F ðz; u; Θ ; tÞ¼ 0:2 þ z; 1 1 3 dt 1þu (1) which validates our embedding method for developing mathe- du 4 ¼F ðz; u; Θ ; tÞ¼ 0:2 þ u: 2 1 3 dt 1þz matical models with multistability. npj Systems Biology and Applications (2022) 10 Published in partnership with the Systems Biology Institute 1234567890():,; S. Wu et al. 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). b The 3D phase portrait of the embedded system (3) with tristability (Three red points: stable steady states; two black points: saddle states). Bistable models for GATA1-PU.1 and GATA-switching modules assumption later based on the experimentally observed mechan- ism. Here the data of the auxiliary variable U is the sum of GATA1 For the two double-negative feedback loops with positive and PU.1. Supplementary Table 4 gives the estimated parameters autoregulation in Fig. 1c, d, we next develop two mathematical of the Z-U module. models for the Z-U module (13) and X-Y module (14). These two An experimental study has identiﬁed GATA2 at chromatin sites models have the same structure but with different model in early-stage erythroblasts , when expression levels of GATA1 parameters. Theorem 1 shows that there are ﬁve possible non- increase as erythropoiesis progresses, GATA1 displaces GATA2 negative equilibria in these models. Theorem 2 indicates that two from chromatin sites. To describe the mechanism of GATA- steady states located on the axis are stable under the given switching, we introduce an additional rate constant k over a time conditions. In addition, Theorem 3 gives the conditions under interval [t , t ] for the displacement rate of GATA2 proteins during which two possible steady states located out of the axis are stable 1 2 the process of GATA-switching, given by (see Methods). We further search for stable steady states of the model with k t 2½t ; t ; 1 2 k ¼ (4) randomly sampled parameters. Supplementary Table 1 gives 0 otherwise : three types of bistable steady states. However, we have not found any parameter samples to realise tristability. To test Since the displacement of GATA2 protein increasing, the robustness properties, we conduct perturbation tests by concentration of GATA1 proteins around the binding site will * * examining the bistable property of the model with slightly increase proportionally to k . Hence, we use rate ψk z for the 53,54 changed model parameters . Our computational results increase of GATA1 during GATA-switching, where ψ is a control demonstrate that a perturbed bistable model with one stable parameter to adjust the availability of GATA1 proteins around steady state located on the axis but another located off the axis chromatin sites. Then the GATA-switching module is modelled by canbefound foramodel withtwo stable steady states located a z dz 1 1 ¼ k z k z; on theaxis(seeSupplementary Table2). Theseresults suggest dt 1þb z 1þb u 1 2 (5) c u du 1 1 that the developed model has very good robustness properties ¼ k u þ ψk z; dt 1þd u 1þd z 1 2 in terms of parameter variations. where z and u are expression levels of GATA2 and GATA1, We next use the approximate Bayesian computation (ABC) 55,56 respectively. Note that the bistability property of this module is rejection algorithm to estimate model parameters based on realised by model (5) using k = 0. Figure 3b gives two simulations the experimental data for erythroiesis and granulopoiesis .We for an unsuccessful switching and a successful switching. It is ﬁrst estimate parameters in the X-Y module that describes assumed that the GATA-switching occurs over the interval [t , t ] regulations between genes GATA1 and PU.1(14). It is assumed 1 2 = [500, 3500]. Simulations show that an adequate displacement of that the prior distribution of each parameter is a uniform GATA2 is the key to achieve GATA-switching using a relatively distribution over the interval [0, 100]. The distance between large value of k 1. experimental data and simulations is measured by ρðX; X Þ¼ ½jx x jþjy y j; Tristable model of the GATA1-GATA2-PU.1 network i i i¼1 After successfully realising the bistability in double-negative feedback loops with positive autoregulation, we next incorpo- where (x , y ) and (x ; y ) are the observed data and simulated data i i i i rate the GATA1-PU.1 regulatory module into the GATA- for genes (X, Y), respectively. Supplementary Table 3 gives the switching module to realise the tristability of HSC differentia- estimated parameters of this module. Figure 3a shows that the tion. We use expression levels of GATA1 in the GATA-switching phase plane of the GATA1-PU.1 sub-system based on estimated module to represent total levels of GATA1 plus PU.1,and embed parameters, which shows that this system is bistable. these two modules together (18)(seeTheorems 4–6in Methods Regarding the Z-U module (13) that describes the regulation of GATA-switching, to be consistent with the module structure, we for more details). The model parameters have the same values ﬁrst assume that GATA1 and GATA2 form a double negative as the corresponding parameters in the Z-U module or the X-Y feedback module with autoregulations, and will modify this module. Supplementary Fig. 1 gives the 3D phase portrait of Published in partnership with the Systems Biology Institute npj Systems Biology and Applications (2022) 10 S. Wu et al. Fig. 3 Realisation of tristability by embedding two bistable sub-systems in hematopoiesis. a Phase plane of the GATA1-PU.1 module showing the bistable property of the proposed model, where A and B are stable steady states; C, D and E are saddle states. b Simulations of GATA-switching of model (5). Upper panel: An unsuccessful switching with a small value of k due to the displacement of GATA2 not being enough for cells to leave the HSCs state (Z state); Lower panel: A successful switching with sufﬁcient displacement of GATA2 by using a large value of k . Cells leave the HSCs state and enter the U state. c The 3D phase portrait of the modiﬁed embedding model (6) with k = 0. Four red points are stable steady states, while the three black points are saddle states. the embedded system, which shows that the embedding 0.7417, 8.6664) of the three genes. In fact, these are exact four model faithfully realises three stable steady states, which also transcriptional states that have been observed in experimental high low high- suggests that the proposed embedding method is a robust studies, namely a PU.1 Gata1/2 state (P1H); a Gata1 low high low approach to develop high order multistable models based on GATA2/PU.1 state (G1H); a Gata2 GATA1/PU.1 state (G2H); bistable models. and a state with low expression of all three genes (LES CMP) . As mentioned in the previous subsection, the GATA-switching Compared with existing modelling studies, our embedding model module is not a perfect double-negative feedback loop. In fact, (6) successfully realises the state with low expression levels of all experimental studies suggest that GATA2 moderately simulates three genes. the expression of gene GATA1 . Thus we make a modiﬁcation to Note that the embedding model is based on the assumption of model (18) by adding the term d z in the ﬁrst equation to GATA-switching, namely the exchange of GATA1 for GATA2 at the represent a weak positive regulation from GATA2 to GATA1.In chromatin site, which controls the expression of genes GATA1 and addition, to avoid zero basal gene expression levels, we add a GATA2. However, a low level of GATA2 at the chromatin site does constant to each equation of the proposed model (18). The not mean the total level of GATA2 in cells is also low. This may be modiﬁed model is given by, the reason for the difference between the simulated state high low α þ α x dx 0 1 1 1 þ d z Gata1 GATA2/PU.1 state (G1H) (namely only GATA1 has high ¼ k x þ ψk z; dt 1 þ β x 1 þ β y 1 þ d z 1 2 2 high- expression) and the experimentally observed state Gata1/2 dy γ þ γ y 1 1 0 1 low ¼ k y; (6) 4 PU.1 state (G1/2H) (namely both GATA1 and GATA2 have high dt 1 þ σ y 1 þ σ x 1 þ d z 1 2 2 dz a0 þ a1z 1 expression levels) . ¼ k z k z; dt 1 þ b z 1 þ b ðx þ yÞ 1 2 where x, y, z represent expression levels of genes GATA1, GATA2 and PU.1, respectively. The values of α , γ , a and d are carefully 0 0 0 Stochastic model for realising heterogeneity selected so that the model simulation still matches experimental Although the modiﬁed embedding model has successfully realised data and the model has at least three stable steady states (see the quad-stability properties, this deterministic model cannot Supplementary Table 5). Figure 3c gives the 3D phase portrait of describe the heterogeneity in the cell fate commitment. Thus, the system (6) with k = 0. Using estimated parameters (see Supple- next question is whether we can use a stochastic model to realise mentary Tables 3–5), the modiﬁed system (6) actually achieves experimental data showing different gene expression levels in quad-stability. In three stable states, one of the three genes has single cells . To answer this question, we propose a stochastic high expression levels but the other two have low expression levels. The fourth stable state has low expression levels (2.3364, differential equations model in Itô form to describe the functions npj Systems Biology and Applications (2022) 10 Published in partnership with the Systems Biology Institute S. Wu et al. Fig. 4 Stochastic simulations showing four stable states that correspond to the experimentally observed four different states. a Simulation of unsuccessful GATA switching that makes the cell stay at the HSC state, which is the G2H state. b Simulation of unsuccessful GATA switching but the cell enters the state with low expression of all three genes, which is the LES CMP state. c Simulation of successful switching that leads to the GMP state with high expression levels of PU.1, which is the P1H state. d Simulation of successful switching that leads to the MEP state with high expression levels of GATA1, which is the G1H state. of noise during the cell lineage speciﬁcation, given by (7) the given k and ψ values. To show the boundary of parameter hi space, we also keep certain sets of parameter values with which α þ α XðtÞ 1 þ d ZðtÞ 1 0 1 1 dXðtÞ¼ k XðtÞþ ψk ZðtÞ dt þ½ω ðk XðtÞþ ψk ZðtÞÞdW ; 3 1 3 1 þ β XðtÞ 1 þ β YðtÞ 1 þ d ZðtÞ t 2 simulations move to one speciﬁc stable state. Figure 5a gives 1 2 hi γ þγ YðtÞ 1 1 2 0 1 proportions of simulations that have successful switching in dYðtÞ¼ k YðtÞ dt þ½ω k YðtÞdW ; 4 2 4 1 þ σ YðtÞ 1 þ σ XðtÞ 1 þ d ZðtÞ t 1 2 2 hi 20,000 simulations. When the value of k is between 0.1 and 0.2, a þa ZðtÞ 0 0 1 1 3 dZðtÞ¼ k ZðtÞ k ZðtÞ dt þ½ω ðk þ k ÞZðtÞdW ; 1 3 1 1 þ b ZðtÞ 1 þ b ðXðtÞþ YðtÞÞ t 1 2 the displacement speed of GATA2 is low, which gives limited relief of negative regulation to PU.1, but GATA1 increases gradually due (7) to GATA-switching and weak positive regulation from GATA2 to 1 2 3 where W , W and W are three independent Wiener processes t t t GATA1. Thus nearly all cells choose the MEP state with high whose increment is a Gaussian random variable ΔW = W(t + Δt) expression levels of GATA1. However, if the value of k is larger, − W(t)~ N(0, Δt), and ω , ω and ω represent noise strengths. The 1 2 3 the negative regulation from GATA2 to PU.1 is eliminated quickly, reason for selecting Itô form is to maintain the mean of the thus the competition between GATA1 and PU.1 will lead cells to stochastic system (7) as the corresponding deterministic system different lineages. When the value of k is relatively large but the (6). To test the inﬂuence of GATA-switching on determining the value of ψ is relatively small, the increase of GATA1 is slow due to transitions between different states, we introduce noise to the smaller value of ψ in GATA-switching. However, the negative coefﬁcient k and consequently to the three degradation regulation from GATA2 to PU.1 declines rapidly due to the larger processes in the model. We use the semi-implicit Euler method value of k . Thus, Fig. 5b shows that the combination of larger k 0 0 to simulate the proposed model . Figure 4 provides four and smaller ψ values allows more cells to move to the GMP stochastic simulations for four different types of cell fate lineage with high expression level of PU.1. If there is no winner in commitments with model parameters k ¼ 0:52, ψ = 0.0005, ω the competition between GATA1 and PU.1, the cell then moves to = 0.04, and ω = ω = 0.08. Figure 4a, b shows two simulations of 2 3 the state with low expression levels of three genes (namely LE3G). unsuccessful GATA switching when the displacement of GATA2 is Figure 5c shows that, when the value of k is larger than 0.2, there not sufﬁcient. However, a sufﬁcient displacement of GATA2 can 0 are four types of simulations as shown in Fig. 5 for a set of k and trigger successful GATA switching, which leads to either the GMP ψ values. We use a MATLAB package to give the violin plot for state with high expression levels of PU.1 in Fig. 4c or the MEP state the expression distributions of three genes in three different with high expression levels of GATA1 in Fig. 4d. cellular states. The violin plot is a combination of a box plot and a To examine the heterogeneity of hematopoiesis with different kernel density plot that illustrates data peaks. The violin plots in displacement rates k and ψ together, we generate 20,000 sto- Fig. 5d match the experimental observations very well . chastic simulations for each set of k and ψ values over the range Regarding the size of basins of attraction, we ﬁrst calculate the of [0.04, 1] and [0, 0.001], respectively. The ranges of k and ψ are determined by numerical testing. If all stochastic simulations distances between the stable states and saddle points in Fig. 3c, move to a single stable state for the given k and ψ values, we which are given in Supplementary Table 6. The minimal distance change the lower bound and/or upper bound of the value range between the G1H state and three saddle points is much larger in order that simulations may move to different stable states for than the minimal distances of the other three stable states to the Published in partnership with the Systems Biology Institute npj Systems Biology and Applications (2022) 10 S. Wu et al. Fig. 5 Distributions of different cell types derived from stochastic simulations. a Frequencies of cells having successful switching for each set of parameters ðk ; ψÞ. b Ratios of GMP cells to MEP cells when cells have successfully switched in a for each set of parameters ðk ; ψÞ. 0 0 c Parameter sets of ðk ; 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 ¼ 0:52 and ψ = 0.0005. saddle points, which suggests that the size of basin of attraction properties. In addition, using cell fate commitments in hemato- for the G1H state is larger than those of the other three stable poiesis as the test problem, we have successfully realised tristability in the GATA-PU.1 module by embedding two bistable states. In addition, we observe the variability of stable states in 20,000 stochastic simulations. Supplementary Table 7 shows that modules together. More importantly, by modifying the model the variations of GATA1 in the G1H state are much larger than using experimentally determined regulatory mechanisms, the developed model successfully realises four stable states that have those of the other two genes when having high expression levels. We also study the relative frequency of LE3G state. Supple- been observed in a recent experimental study . mentary Fig. 2 shows that, for a ﬁxed value of parameter ψ, the In this study the stable states are achieved by a model without frequency increases as the value of k increases. In addition, for a high cooperativity (i.e. Hill coefﬁcient n = 1). Recently, the dynamics of toggle triad with self-activations have attracted ﬁxed value of k , the frequency decreases as the value of ψ 60,61 increases. The variation of parameter ψ is much more important much attention . Mathematical models with high cooperativity than that of parameter k . For the simulations showing in Fig. 5d, have been developed to achieve pentastable, namely a hybrid X/Y state with high X, high Y and low Z. We tried to realise the frequency is 0.1080 with k ¼ 0:52 and ψ = 0.0005. Figure 5d pentastability by using our proposed model with high coopera- and Supplementary Fig. 2 suggest that more cells remain in the LE3G or P1H (GMP) state if GATA2 leaves the chromatin site fast tivity (n = 2 or 3), but numerical tests were not successful. Thus, high cooperativity in self-activation may be essential to realise (i.e. a large k value) and the expression of GATA1 is slow (i.e. a pentastable. This is an interesting problem that will be the topic of 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 further studies. Despite the assumption of a binary choice in each sub-module, frequency of the LE3G state is low, which is consistent with the the developed model is able to realise a rich variety of dynamics. results in a recent study . Our research suggests that, depending on the properties of bistable systems, the embedding model of two bistable modules DISCUSSIONS may have more than three stable steady states. In addition, using Inspired by Waddington’s epigenetic landscape model, we the embedding method in Fig. 1,the state U is not a meta-stable assume that a multistable system makes a series of binary state but actually disappears from the system. Simulations show decisions for the selection of multiple evolutionary pathways. that, when the system leaves the high GATA2 expression state Compared with modelling studies for multistable networks, it is due to GATA-switching, genes GATA1 and PU.1 begin to increase relatively easy to develop models with bistability and there is a their expression levels. Each stochastic simulation will reach one rich literature for studying bistable networks. Thus, our proposed of the steady states with either high GATA1 levels or high PU.1 embedding method is an effective approach to develop multi- levels or return to the stem cell state. These simulations are stable models based on well-studied models with bistable consistent with the CLOUD-HSPC model in which differentiation npj Systems Biology and Applications (2022) 10 Published in partnership with the Systems Biology Institute S. Wu et al. is a process of uncommitted cells in transitory states that following model to describe the dynamics, given by 62–64 gradually acquire uni-lineage priming . In addition, stochastic dz a z 1 ¼F ðz; u; Θ ; tÞ¼ k z; 1 1 1 dt 1þb z 1þb u 1 2 simulations demonstrate that noise plays a key role in (13) c u du 1 1 ¼F ðz; u; Θ ; tÞ¼ k u: 2 1 2 determining different differentiation pathways. dt 1þd u 1þd z 1 2 This work uses differential equation models to determine stable Similarly, based on the formalism (9) with Y = {x, y} and Θ = {α , β , β , γ , 2 1 1 2 1 steady states and then employs corresponding stochastic models σ , σ , k , k }, the dynamics of the X− Y subsystem is modelled by 1 2 3 4 to realise the functions of noise. However, experimental studies α x dx 1 1 ¼G ðx; y; Θ ; tÞ¼ k x; 1 2 3 dt 1þβ x 1þβ y have shown that gene expression is a bursting process. The 1 2 (14) dy γ y 1 1 challenge is how to determine conditions for realising the ¼G ðx; y; Θ ; tÞ¼ k y; 2 2 4 dt 1þσ y 1þσ x 1 2 multistable properties in stochastic models with bursting pro- where x and y are expression levels of genes X and Y, respectively; α and cesses. In addition, hematopoiesis is a process to produce all γ represent expression rates; β , β , σ and σ represent association rates 1 1 2 1 2 mature blood cells. This is an ideal test system to develop of corresponding proteins to binding-sites; and k and k are self- 3 4 mathematical models with multistable dynamics. An interesting degradation rates. The model of the Z-U subsystem has the same structure question is how to embed more modules with more transcription but may have different values of model parameters. To obtain the factors to develop mathematical models with more stable steady bistability, we establish the following theorems for our proposed models for these two sub-systems. Since they have the same structure, we only states. All these issues will be interesting topics of further research. give the theorems for the X-Y sub-system. Theorem 1. There are at most ﬁve sets of non-negative equilibria for METHODS model (14). Embedding method to couple models together α k 1 3 We propose a framework to model regulatory networks with multiple 1. There are three equilibria: (0, 0), (x , 0) and (0, y ), where x ¼ e e e k β γ k 1 4 stable steady states based on the embedding of sub-systems with less and y ¼ ,if α > k and γ > k . 1 3 1 4 e k σ 4 1 B C stable steady states. It is assumed that we need to study a regulatory 2. There are two other equilibria: ðx ; y Þ and ðx ; y Þ.If > 0, > 0 1 1 2 2 A A network that consists of two regulatory modules. The ﬁrst module has and B 4AC 0, then x and x are positive real solutions of the 1 2 following equation, genes X , and it is modelled by the following equation Am þBm þC ¼ 0; (15) dX (8) ¼F ðX ; X ; ; X ; X ; ; X ; Θ ; tÞ i 1 2 n nþ1 nþN 1 dt where m ¼ β x;A¼ A B B ;B¼ A B 1 þ A B A B þ A B , 1 1 1 1 1 1 1 1 2 2 1 β γ α σ 2 1 2 1 C¼ A þ A 1 A B , A ¼ ; A ¼ ; B ¼ and B ¼ . 1 2 1 2 1 2 1 2 for i = 1, 2, ⋯ , n + N, where Θ includes model parameters of F . The σ k β k 1 i 1 3 1 4 3. To have positive values of y and y , the following conditions should 1 2 second module has the following model be satisﬁed, dY ¼G ðY ; Y ; ; Y ; Θ ; tÞ (9) j 1 2 m 2 A 1 B 1 2 2 dt x < or x < : (16) 1;2 1;2 β σ 1 2 for j = 1, 2, ⋯ , m, where Θ includes model parameters of G . In these two 2 j models, FðX; Θ ; tÞ and GðY; Θ ; tÞ are non-linear vector ﬁelds. To develop 1 2 Moreover, to study the bistability, it is necessary to establish conditions mathematical models with more stable steady states, we propose an of stability/instability for each equilibrium state. We ﬁrst give the following embedding method by assuming that X (k = 1, .. ., N) are functions of n+k conditions for each equilibrium state that locates on an axis. variables Y , Y , ⋯ , Y , given by 1 2 m Theorem 2. The X-Y system has three equilibria: (0, 0), (x , 0) and (0, y ). X ¼H ðY ; Y ; ; Y Þ: (10) e e nþk k 1 2 m 1. The equilibrium state (0, 0) is unstable if α > k and γ > k . In this way, we obtain an embedding system 1 3 1 4 2. The equilibrium state (x , 0) is stable if <k . e 4 1þσ x 2 e dW 3. The equilibrium state (0, y ) is stable if <k . e 3 1þβ y (11) ¼ FðW; Θ ; tÞ; 2 e dt In addition, we give the following stable conditions for each equilibrium where W = (X , X , ⋯ , X , Y , Y , ⋯ , Y ) represents all genes in the system, 1 2 n 1 2 m state that locates within the 2-dimensional positive real space. F denotes the embedding system from two modules with gene X and Y i i with function H . In addition, Θ = Θ ∪ Θ is the model parameters space. k 1 2 Theorem 3. The positive equilibria ðx ; y Þ and ðx ; y Þ are stable if the This embedding system (11) consists of two components: 1 1 2 2 following condition is satisﬁed. dX ¼F ðX ; X ; ; X ; H ðY ; Y ; ; Y Þ; Θ ; tÞ; i 1 2 n k 1 2 m dt β σ η ξ β σ θ ρ > 0: (17) (12) 1 2 x 1 y x 2 y dY ¼R ðX ; X ; ; X ; Y ; Y ; ; Y ; Θ ; tÞ j 1 2 n 1 2 m dt where θ = 1 + β x, η = 1 + β y, ρ = 1 + σ y and ξ = 1 + σ x. x 1 y 2 y 1 x 2 for i = 1, 2, ⋯ , n, k = 1, .. ., N and j = 1, 2, ⋯ , m. Since each X is regulated by the X (k = 1, .. ., N), and X are functions of Y , Y , ⋯ , Y , the In summary, Theorem 1 gives the existence conditions of the equilibria n+k n+k 1 2 m expressions of each gene Y is also regulated by X (i = 1, .. ., n). The non- for our proposed two-node systems. Theorems 2 and 3 provide the j i linear vector ﬁeld GðY; Θ ; tÞ in Eq. (9) will then be transformed into a new necessary conditions for stability properties of these equilibria. According non-linear vector ﬁeld RðW; Θ ; tÞ, which includes both genes X and Y to these theorems, we can easily check whether two-node systems have i i from two sub-systems with their corresponding regulations. Note that this bistability based on generated samples of model parameters. The proofs of is a general idea to develop mathematical models with more stable steady these theorems are given in Supplementary Notes. states. Depending on the speciﬁc formalism and properties of sub-systems, the embedding system may have different results regarding multiple Perturbation analysis of bistable models stable steady states with different conditions. In this study, we only focus We have proved that systems (13)and (14) have bistable steady states on the systems with Shea-Ackers formalism . under the conditions in Theorems 2 or 3. Next we use the random search method to ﬁnd the model parameters with which the system has bistable Model development for bistability properties steady states. We ﬁrst generate a sample for each model parameter from We ﬁrst develop a model for the network in Fig. 1c with bistability the uniform distribution over the interval [0, A] and then test whether the properties. Suppose that two sub-systems, namely the Z-U system and X-Y system with the sampled parameters satisﬁes the conditions in Theorems sub-system, have the same structure of a double-negative feedback loop 2 or 3. If the conditions are satisﬁed, we solve nonlinear equations of the and positive autoregulations. For the Z-U system, based on the formalism system to ﬁnd the steady states. We test different values of A and ﬁnd (8) with X = {z, u} and Θ = {a , b , b , c , d , d , k , k }, we propose the that the system has bistable steady states when A = 10. To ﬁnd more 1 1 1 2 1 1 2 1 2 Published in partnership with the Systems Biology Institute npj Systems Biology and Applications (2022) 10 S. Wu et al. types of bistable states, we test 10000 sets of parameters from the DATA AVAILABILITY uniform distribution over the interval [0, 10]. Supplementary Table 1 gives No datasets were generated during the current study. The experimental data for three types of bistable steady states, namely Case 1: (x , 0) and (0, y ); e e erythroiesis and granulopoiesis that support the parameter estimation of this study Case 2: (x ,0) and ðx ; y Þ; and Case 3: (0, y )and ðx ; y Þ: All stable states e e are available in the published paper at https://www.nature.com/articles/s41586-020- 1 1 2 2 in case 1 are located on the coordinate axis. We add a perturbation to 2432-4 . each estimated coefﬁcient c as c = [ε ×(P − 0.5) + 1] × c,where P is a uniformly distributed random variable over the interval [0, 1], and ε is the strength of perturbation. Supplementary Table 2 shows that the two CODE AVAILABILITY other cases of bistability can be obtained by the perturbed coefﬁcients The code used to perform the analyses presented in the current study is available from Case 1. from the corresponding author on reasonable request. 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Cell fate decision as high-dimensional critical state transition. PLoS Biol. 14,1–28 (2016). ADDITIONAL INFORMATION 44. Semrau, S. et al. Dynamics of lineage commitment revealed by single-cell tran- Supplementary information The online version contains supplementary material scriptomics of differentiating embryonic stem cells. Nat. Commun. 8, 1096 (2017). available at https://doi.org/10.1038/s41540-022-00220-1. 45. Mackey, M. C. Periodic hematological disorders: Quintessential examples of dynamical diseases. Chaos 30, 063123 (2020). Correspondence and requests for materials should be addressed to Tianhai Tian. 46. Tian, T. & Burrage, K. Stochastic models for regulatory networks of the genetic toggle switch. Proc. Natl. Acad. Sci. USA 103, 8372–8377 (2006). Reprints and permission information is available at http://www.nature.com/ 47. Feng, J., Kessler, D. A., Ben-Jacob, E. & Levine, H. Growth feedback as a basis for reprints persister bistability. Proc. Natl. Acad. Sci. 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Publisher: Springer Journals
Copyright: Copyright © The Author(s) 2022
eISSN: 2056-7189
DOI: 10.1038/s41540-022-00220-1
Publisher site: See Article on Publisher Site

Abstract

www.nature.com/npjsba ARTICLE OPEN A robust method for designing multistable systems by embedding bistable subsystems 1 2 1✉ Siyuan Wu , Tianshou Zhou and Tianhai Tian 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 together. Using the GATA1-GATA2- PU.1 module in hematopoiesis as the test system, we ﬁrst develop a tristable model based on two bistable models without any high cooperative coefﬁcients, and then modify the tristable model based on experimentally determined mechanisms. The modiﬁed model successfully realises four stable steady states and accurately reﬂects a recent experimental observation showing four transcriptional states. In addition, we develop a stochastic model, and stochastic simulations successfully realise the experimental observations in single cells. These results suggest that the proposed method is a general approach to develop mathematical models for realising multistability and heterogeneity in complex systems. npj Systems Biology and Applications (2022) 8:10 ; https://doi.org/10.1038/s41540-022-00220-1 INTRODUCTION suggested that GATA2 and GATA1 sequentially bind the same 29,30 cis-elements, which is referred to as the GATA-switching . Multistability is the characteristic of a system that exhibits two or Mathematical modelling is a powerful tool to accurately more mutually exclusive stable states. This phenomenon has been describe the dynamics of hematopoiesis and to explore the observed in many different disciplines of science, including 1–4 5–8 regulatory mechanisms for controlling the transitions between genetic regulatory networks , cell signalling pathways , meta- 31–37 9 10,11 12 different cell types . For the GATA1-PU.1 module, Hill equations bolic networks , ecosystems , neuroscience , laser sys- 13,14 15 with high cooperativity were initially used to realise tristability .In tems , and quantum systems . When external and/or addition, mathematical models have been proposed to achieve internal conditions change, the system may switch from one bistability in gene regulatory networks without any high steady state to another either randomly by perturbations or in a 39,40 cooperativity coefﬁcients . Bifurcation theory is also an efﬁcient desired way according to the control strategies. In recent years method to explore the mechanisms of GATA1-PU.1 module .We mathematical models with multistability have been developed for have proposed a mathematical model to realise the mechanisms theoretical analysis and computer simulations, which shed light on of GATA-switching and designed an effective algorithm to realise the mechanisms that generate multistability and control the 16–19 tristability of mathematical models . Moreover, the underlying transition between steady states . mechanisms of how the stem/progenitor cells leave the stable As one of the important molecular systems showing multi- steady states and commit to a speciﬁc lineage were also revealed stability, hematopoiesis is a highly integrated developmental with the assistance of mathematical models . At the single cell process that controls the proliferation, differentiation and 20,21 level, the differentiation processes of embryonic stem cells were maturation of hematopoietic stem cells (HSCs) . HSCs have simulated by Langevin equations, which helped to identify the features of self-renewal and multipotency as well as the ability to differentiate into multipotent progenitors (MPPs). Each of these potential transcriptional regulators of lineage decision and commitment . Mathematical models have also been used to cell types is regarded as a stable state of the multistable system. In study the dynamical properties of diseases such as periodic addition, the formation of white and red blood cells is a dynamical haematological disorders . process that transits a cell from one stable cell type to another. Although these attempts have realised tristability by using This process begins with the differentiation of HSCs and enters the different assumptions, it is still a challenge to develop mathema- main stage at which cells reach either common myeloid 22,23 tical models to realise tristability using both the realistic regulatory progenitors (CMPs) or common lymphoid progenitors (CLPs) . Transcription factors play a key role in controlling the process of mechanisms and experimental data. On the other hand, blood cell lineage speciﬁcation. Experimental studies have substantial research studies have been conducted to develop 3,46–51 demonstrated that the genetic module GATA1-PU.1 is a vital mathematical models for realising bistability properties . Thus, the question is whether we can develop mathematical component for the fate commitment of CMPs between erythro- 24,25 models with tristability or higher order of multistability by using poiesis and granulopoiesis . HSCs are more likely to choose megakaryocyte/erythroid progenitors (MEPs) with high expression bistable models. To address this issue, we propose a robust levels of GATA1 , or conversely to choose granulocyte/macro- method to develop multistable models by embedding bistable phage progenitors (GMPs) with high expression levels of PU.1 .In models together. Using the GATA1-GATA2-PU.1 module as a addition, the regulation between genes GATA1 and GATA2 is an testing model, we develop a tristable model based on two essential driver of hematopoiesis . Experimental studies systems that have no high cooperativity coefﬁcients. 1 2 School of Mathematics, Monash University, Melbourne, VIC, Australia. School of Mathematics and Statistics, Sun Yet-Sen University, Guangzhou, China. email: Tianhai.Tian@monash.edu Published in partnership with the Systems Biology Institute 1234567890():,; S. Wu et al. Fig. 1 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. RESULTS It is assumed that the ﬁrst Z-U module follows model (1) and the second X-Y module satisﬁes the same model with same parameter Embedding method for designing multistable models space Θ , but different variables x and y, given by The motivation of this work is to develop a mathematical model to dx 4 realise the tristable property of the HSC genetic regulatory network ¼G ðx; y; Θ ; tÞ¼ 0:2 þ x; 1 1 3 dt 1þy (2) in Fig. 1a based on experimental observations. Figure 1b, e dy ¼G ðx; y; Θ ; tÞ¼ 0:2 þ y: 2 1 3 dt 1þx illustrates the embedding method to couple two bistable modules in a network together, where ’→ ’ and ’⊣’ denote the activating Now we embed these two sub-systems together using and inhibiting regulations, respectively. Variable U in the ﬁrst Z-U u ¼Hðx; yÞ¼ x þ y. Since gene z is negatively regulated by gene module is an auxiliary node, which is assumed to be U= μX + δY, u in the sub-system (1), and u is a function of genes x and y, the where μ and δ are two positive parameters. When the system stays expressions of genes x and y are also negatively regulated by gene in the state with a high expression level of Z and a low level of U, z in the new embedding model. Then the non-linear vector ﬁelds the expression levels of X and Y are low. However, when the G ðx; y; Θ ; tÞ are transformed into new non-linear vector ﬁelds 1;2 1 R ðx; y; z; Θ ; tÞ, respectively, which include genes x, y and z system has a low expression level of Z and a high level of U, the 1;2 1 from two sub-systems with negative regulations from gene z to system triggers the second module X-Y to choose either a high genes x and y. Therefore, the new model with three variables is level of X and a low level of Y or a low level of X and a high level of given by Y. In this way we realise the system with three stable states in dx 4 which one of the three variables (namely Z, X or Y) is at the high ¼R ðx; y; z; Θ ; tÞ¼ 0:2 þ x; 1 1 3 3 dt ð1þy Þð1þz Þ expression state but the other two are at low expression states. dy ¼R ðx; y; z; Θ ; tÞ¼ 0:2 þ y; w 1 3 3 (3) To demonstrate the effectiveness of the proposed embedding dt ð1þx Þð1þz Þ method, we use the toggle switch network as the test system . dz 4 ¼F ðz; u ¼ x þ y; Θ ; tÞ¼ 0:2 þ z: 1 1 3 dt 1þðxþyÞ This network consists of two genes that form a double negative feedback loop and is modelled by the following equations with Figure 2a shows the phase plane of the toggle switch sub-system parameter space Θ = {a = 0.2, b = 4, c = 3}, given by (1) with bistability properties, and Fig. 2b provides the 3D phase portrait of the embedded model (3) with three stable steady dz 4 states. The embedded model successfully realised the tristability, ¼F ðz; u; Θ ; tÞ¼ 0:2 þ z; 1 1 3 dt 1þu (1) which validates our embedding method for developing mathe- du 4 ¼F ðz; u; Θ ; tÞ¼ 0:2 þ u: 2 1 3 dt 1þz matical models with multistability. npj Systems Biology and Applications (2022) 10 Published in partnership with the Systems Biology Institute 1234567890():,; S. Wu et al. 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). b The 3D phase portrait of the embedded system (3) with tristability (Three red points: stable steady states; two black points: saddle states). Bistable models for GATA1-PU.1 and GATA-switching modules assumption later based on the experimentally observed mechan- ism. Here the data of the auxiliary variable U is the sum of GATA1 For the two double-negative feedback loops with positive and PU.1. Supplementary Table 4 gives the estimated parameters autoregulation in Fig. 1c, d, we next develop two mathematical of the Z-U module. models for the Z-U module (13) and X-Y module (14). These two An experimental study has identiﬁed GATA2 at chromatin sites models have the same structure but with different model in early-stage erythroblasts , when expression levels of GATA1 parameters. Theorem 1 shows that there are ﬁve possible non- increase as erythropoiesis progresses, GATA1 displaces GATA2 negative equilibria in these models. Theorem 2 indicates that two from chromatin sites. To describe the mechanism of GATA- steady states located on the axis are stable under the given switching, we introduce an additional rate constant k over a time conditions. In addition, Theorem 3 gives the conditions under interval [t , t ] for the displacement rate of GATA2 proteins during which two possible steady states located out of the axis are stable 1 2 the process of GATA-switching, given by (see Methods). We further search for stable steady states of the model with k t 2½t ; t ; 1 2 k ¼ (4) randomly sampled parameters. Supplementary Table 1 gives 0 otherwise : three types of bistable steady states. However, we have not found any parameter samples to realise tristability. To test Since the displacement of GATA2 protein increasing, the robustness properties, we conduct perturbation tests by concentration of GATA1 proteins around the binding site will * * examining the bistable property of the model with slightly increase proportionally to k . Hence, we use rate ψk z for the 53,54 changed model parameters . Our computational results increase of GATA1 during GATA-switching, where ψ is a control demonstrate that a perturbed bistable model with one stable parameter to adjust the availability of GATA1 proteins around steady state located on the axis but another located off the axis chromatin sites. Then the GATA-switching module is modelled by canbefound foramodel withtwo stable steady states located a z dz 1 1 ¼ k z k z; on theaxis(seeSupplementary Table2). Theseresults suggest dt 1þb z 1þb u 1 2 (5) c u du 1 1 that the developed model has very good robustness properties ¼ k u þ ψk z; dt 1þd u 1þd z 1 2 in terms of parameter variations. where z and u are expression levels of GATA2 and GATA1, We next use the approximate Bayesian computation (ABC) 55,56 respectively. Note that the bistability property of this module is rejection algorithm to estimate model parameters based on realised by model (5) using k = 0. Figure 3b gives two simulations the experimental data for erythroiesis and granulopoiesis .We for an unsuccessful switching and a successful switching. It is ﬁrst estimate parameters in the X-Y module that describes assumed that the GATA-switching occurs over the interval [t , t ] regulations between genes GATA1 and PU.1(14). It is assumed 1 2 = [500, 3500]. Simulations show that an adequate displacement of that the prior distribution of each parameter is a uniform GATA2 is the key to achieve GATA-switching using a relatively distribution over the interval [0, 100]. The distance between large value of k 1. experimental data and simulations is measured by ρðX; X Þ¼ ½jx x jþjy y j; Tristable model of the GATA1-GATA2-PU.1 network i i i¼1 After successfully realising the bistability in double-negative feedback loops with positive autoregulation, we next incorpo- where (x , y ) and (x ; y ) are the observed data and simulated data i i i i rate the GATA1-PU.1 regulatory module into the GATA- for genes (X, Y), respectively. Supplementary Table 3 gives the switching module to realise the tristability of HSC differentia- estimated parameters of this module. Figure 3a shows that the tion. We use expression levels of GATA1 in the GATA-switching phase plane of the GATA1-PU.1 sub-system based on estimated module to represent total levels of GATA1 plus PU.1,and embed parameters, which shows that this system is bistable. these two modules together (18)(seeTheorems 4–6in Methods Regarding the Z-U module (13) that describes the regulation of GATA-switching, to be consistent with the module structure, we for more details). The model parameters have the same values ﬁrst assume that GATA1 and GATA2 form a double negative as the corresponding parameters in the Z-U module or the X-Y feedback module with autoregulations, and will modify this module. Supplementary Fig. 1 gives the 3D phase portrait of Published in partnership with the Systems Biology Institute npj Systems Biology and Applications (2022) 10 S. Wu et al. Fig. 3 Realisation of tristability by embedding two bistable sub-systems in hematopoiesis. a Phase plane of the GATA1-PU.1 module showing the bistable property of the proposed model, where A and B are stable steady states; C, D and E are saddle states. b Simulations of GATA-switching of model (5). Upper panel: An unsuccessful switching with a small value of k due to the displacement of GATA2 not being enough for cells to leave the HSCs state (Z state); Lower panel: A successful switching with sufﬁcient displacement of GATA2 by using a large value of k . Cells leave the HSCs state and enter the U state. c The 3D phase portrait of the modiﬁed embedding model (6) with k = 0. Four red points are stable steady states, while the three black points are saddle states. the embedded system, which shows that the embedding 0.7417, 8.6664) of the three genes. In fact, these are exact four model faithfully realises three stable steady states, which also transcriptional states that have been observed in experimental high low high- suggests that the proposed embedding method is a robust studies, namely a PU.1 Gata1/2 state (P1H); a Gata1 low high low approach to develop high order multistable models based on GATA2/PU.1 state (G1H); a Gata2 GATA1/PU.1 state (G2H); bistable models. and a state with low expression of all three genes (LES CMP) . As mentioned in the previous subsection, the GATA-switching Compared with existing modelling studies, our embedding model module is not a perfect double-negative feedback loop. In fact, (6) successfully realises the state with low expression levels of all experimental studies suggest that GATA2 moderately simulates three genes. the expression of gene GATA1 . Thus we make a modiﬁcation to Note that the embedding model is based on the assumption of model (18) by adding the term d z in the ﬁrst equation to GATA-switching, namely the exchange of GATA1 for GATA2 at the represent a weak positive regulation from GATA2 to GATA1.In chromatin site, which controls the expression of genes GATA1 and addition, to avoid zero basal gene expression levels, we add a GATA2. However, a low level of GATA2 at the chromatin site does constant to each equation of the proposed model (18). The not mean the total level of GATA2 in cells is also low. This may be modiﬁed model is given by, the reason for the difference between the simulated state high low α þ α x dx 0 1 1 1 þ d z Gata1 GATA2/PU.1 state (G1H) (namely only GATA1 has high ¼ k x þ ψk z; dt 1 þ β x 1 þ β y 1 þ d z 1 2 2 high- expression) and the experimentally observed state Gata1/2 dy γ þ γ y 1 1 0 1 low ¼ k y; (6) 4 PU.1 state (G1/2H) (namely both GATA1 and GATA2 have high dt 1 þ σ y 1 þ σ x 1 þ d z 1 2 2 dz a0 þ a1z 1 expression levels) . ¼ k z k z; dt 1 þ b z 1 þ b ðx þ yÞ 1 2 where x, y, z represent expression levels of genes GATA1, GATA2 and PU.1, respectively. The values of α , γ , a and d are carefully 0 0 0 Stochastic model for realising heterogeneity selected so that the model simulation still matches experimental Although the modiﬁed embedding model has successfully realised data and the model has at least three stable steady states (see the quad-stability properties, this deterministic model cannot Supplementary Table 5). Figure 3c gives the 3D phase portrait of describe the heterogeneity in the cell fate commitment. Thus, the system (6) with k = 0. Using estimated parameters (see Supple- next question is whether we can use a stochastic model to realise mentary Tables 3–5), the modiﬁed system (6) actually achieves experimental data showing different gene expression levels in quad-stability. In three stable states, one of the three genes has single cells . To answer this question, we propose a stochastic high expression levels but the other two have low expression levels. The fourth stable state has low expression levels (2.3364, differential equations model in Itô form to describe the functions npj Systems Biology and Applications (2022) 10 Published in partnership with the Systems Biology Institute S. Wu et al. Fig. 4 Stochastic simulations showing four stable states that correspond to the experimentally observed four different states. a Simulation of unsuccessful GATA switching that makes the cell stay at the HSC state, which is the G2H state. b Simulation of unsuccessful GATA switching but the cell enters the state with low expression of all three genes, which is the LES CMP state. c Simulation of successful switching that leads to the GMP state with high expression levels of PU.1, which is the P1H state. d Simulation of successful switching that leads to the MEP state with high expression levels of GATA1, which is the G1H state. of noise during the cell lineage speciﬁcation, given by (7) the given k and ψ values. To show the boundary of parameter hi space, we also keep certain sets of parameter values with which α þ α XðtÞ 1 þ d ZðtÞ 1 0 1 1 dXðtÞ¼ k XðtÞþ ψk ZðtÞ dt þ½ω ðk XðtÞþ ψk ZðtÞÞdW ; 3 1 3 1 þ β XðtÞ 1 þ β YðtÞ 1 þ d ZðtÞ t 2 simulations move to one speciﬁc stable state. Figure 5a gives 1 2 hi γ þγ YðtÞ 1 1 2 0 1 proportions of simulations that have successful switching in dYðtÞ¼ k YðtÞ dt þ½ω k YðtÞdW ; 4 2 4 1 þ σ YðtÞ 1 þ σ XðtÞ 1 þ d ZðtÞ t 1 2 2 hi 20,000 simulations. When the value of k is between 0.1 and 0.2, a þa ZðtÞ 0 0 1 1 3 dZðtÞ¼ k ZðtÞ k ZðtÞ dt þ½ω ðk þ k ÞZðtÞdW ; 1 3 1 1 þ b ZðtÞ 1 þ b ðXðtÞþ YðtÞÞ t 1 2 the displacement speed of GATA2 is low, which gives limited relief of negative regulation to PU.1, but GATA1 increases gradually due (7) to GATA-switching and weak positive regulation from GATA2 to 1 2 3 where W , W and W are three independent Wiener processes t t t GATA1. Thus nearly all cells choose the MEP state with high whose increment is a Gaussian random variable ΔW = W(t + Δt) expression levels of GATA1. However, if the value of k is larger, − W(t)~ N(0, Δt), and ω , ω and ω represent noise strengths. The 1 2 3 the negative regulation from GATA2 to PU.1 is eliminated quickly, reason for selecting Itô form is to maintain the mean of the thus the competition between GATA1 and PU.1 will lead cells to stochastic system (7) as the corresponding deterministic system different lineages. When the value of k is relatively large but the (6). To test the inﬂuence of GATA-switching on determining the value of ψ is relatively small, the increase of GATA1 is slow due to transitions between different states, we introduce noise to the smaller value of ψ in GATA-switching. However, the negative coefﬁcient k and consequently to the three degradation regulation from GATA2 to PU.1 declines rapidly due to the larger processes in the model. We use the semi-implicit Euler method value of k . Thus, Fig. 5b shows that the combination of larger k 0 0 to simulate the proposed model . Figure 4 provides four and smaller ψ values allows more cells to move to the GMP stochastic simulations for four different types of cell fate lineage with high expression level of PU.1. If there is no winner in commitments with model parameters k ¼ 0:52, ψ = 0.0005, ω the competition between GATA1 and PU.1, the cell then moves to = 0.04, and ω = ω = 0.08. Figure 4a, b shows two simulations of 2 3 the state with low expression levels of three genes (namely LE3G). unsuccessful GATA switching when the displacement of GATA2 is Figure 5c shows that, when the value of k is larger than 0.2, there not sufﬁcient. However, a sufﬁcient displacement of GATA2 can 0 are four types of simulations as shown in Fig. 5 for a set of k and trigger successful GATA switching, which leads to either the GMP ψ values. We use a MATLAB package to give the violin plot for state with high expression levels of PU.1 in Fig. 4c or the MEP state the expression distributions of three genes in three different with high expression levels of GATA1 in Fig. 4d. cellular states. The violin plot is a combination of a box plot and a To examine the heterogeneity of hematopoiesis with different kernel density plot that illustrates data peaks. The violin plots in displacement rates k and ψ together, we generate 20,000 sto- Fig. 5d match the experimental observations very well . chastic simulations for each set of k and ψ values over the range Regarding the size of basins of attraction, we ﬁrst calculate the of [0.04, 1] and [0, 0.001], respectively. The ranges of k and ψ are determined by numerical testing. If all stochastic simulations distances between the stable states and saddle points in Fig. 3c, move to a single stable state for the given k and ψ values, we which are given in Supplementary Table 6. The minimal distance change the lower bound and/or upper bound of the value range between the G1H state and three saddle points is much larger in order that simulations may move to different stable states for than the minimal distances of the other three stable states to the Published in partnership with the Systems Biology Institute npj Systems Biology and Applications (2022) 10 S. Wu et al. Fig. 5 Distributions of different cell types derived from stochastic simulations. a Frequencies of cells having successful switching for each set of parameters ðk ; ψÞ. b Ratios of GMP cells to MEP cells when cells have successfully switched in a for each set of parameters ðk ; ψÞ. 0 0 c Parameter sets of ðk ; 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 ¼ 0:52 and ψ = 0.0005. saddle points, which suggests that the size of basin of attraction properties. In addition, using cell fate commitments in hemato- for the G1H state is larger than those of the other three stable poiesis as the test problem, we have successfully realised tristability in the GATA-PU.1 module by embedding two bistable states. In addition, we observe the variability of stable states in 20,000 stochastic simulations. Supplementary Table 7 shows that modules together. More importantly, by modifying the model the variations of GATA1 in the G1H state are much larger than using experimentally determined regulatory mechanisms, the developed model successfully realises four stable states that have those of the other two genes when having high expression levels. We also study the relative frequency of LE3G state. Supple- been observed in a recent experimental study . mentary Fig. 2 shows that, for a ﬁxed value of parameter ψ, the In this study the stable states are achieved by a model without frequency increases as the value of k increases. In addition, for a high cooperativity (i.e. Hill coefﬁcient n = 1). Recently, the dynamics of toggle triad with self-activations have attracted ﬁxed value of k , the frequency decreases as the value of ψ 60,61 increases. The variation of parameter ψ is much more important much attention . Mathematical models with high cooperativity than that of parameter k . For the simulations showing in Fig. 5d, have been developed to achieve pentastable, namely a hybrid X/Y state with high X, high Y and low Z. We tried to realise the frequency is 0.1080 with k ¼ 0:52 and ψ = 0.0005. Figure 5d pentastability by using our proposed model with high coopera- and Supplementary Fig. 2 suggest that more cells remain in the LE3G or P1H (GMP) state if GATA2 leaves the chromatin site fast tivity (n = 2 or 3), but numerical tests were not successful. Thus, high cooperativity in self-activation may be essential to realise (i.e. a large k value) and the expression of GATA1 is slow (i.e. a pentastable. This is an interesting problem that will be the topic of 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 further studies. Despite the assumption of a binary choice in each sub-module, frequency of the LE3G state is low, which is consistent with the the developed model is able to realise a rich variety of dynamics. results in a recent study . Our research suggests that, depending on the properties of bistable systems, the embedding model of two bistable modules DISCUSSIONS may have more than three stable steady states. In addition, using Inspired by Waddington’s epigenetic landscape model, we the embedding method in Fig. 1,the state U is not a meta-stable assume that a multistable system makes a series of binary state but actually disappears from the system. Simulations show decisions for the selection of multiple evolutionary pathways. that, when the system leaves the high GATA2 expression state Compared with modelling studies for multistable networks, it is due to GATA-switching, genes GATA1 and PU.1 begin to increase relatively easy to develop models with bistability and there is a their expression levels. Each stochastic simulation will reach one rich literature for studying bistable networks. Thus, our proposed of the steady states with either high GATA1 levels or high PU.1 embedding method is an effective approach to develop multi- levels or return to the stem cell state. These simulations are stable models based on well-studied models with bistable consistent with the CLOUD-HSPC model in which differentiation npj Systems Biology and Applications (2022) 10 Published in partnership with the Systems Biology Institute S. Wu et al. is a process of uncommitted cells in transitory states that following model to describe the dynamics, given by 62–64 gradually acquire uni-lineage priming . In addition, stochastic dz a z 1 ¼F ðz; u; Θ ; tÞ¼ k z; 1 1 1 dt 1þb z 1þb u 1 2 simulations demonstrate that noise plays a key role in (13) c u du 1 1 ¼F ðz; u; Θ ; tÞ¼ k u: 2 1 2 determining different differentiation pathways. dt 1þd u 1þd z 1 2 This work uses differential equation models to determine stable Similarly, based on the formalism (9) with Y = {x, y} and Θ = {α , β , β , γ , 2 1 1 2 1 steady states and then employs corresponding stochastic models σ , σ , k , k }, the dynamics of the X− Y subsystem is modelled by 1 2 3 4 to realise the functions of noise. However, experimental studies α x dx 1 1 ¼G ðx; y; Θ ; tÞ¼ k x; 1 2 3 dt 1þβ x 1þβ y have shown that gene expression is a bursting process. The 1 2 (14) dy γ y 1 1 challenge is how to determine conditions for realising the ¼G ðx; y; Θ ; tÞ¼ k y; 2 2 4 dt 1þσ y 1þσ x 1 2 multistable properties in stochastic models with bursting pro- where x and y are expression levels of genes X and Y, respectively; α and cesses. In addition, hematopoiesis is a process to produce all γ represent expression rates; β , β , σ and σ represent association rates 1 1 2 1 2 mature blood cells. This is an ideal test system to develop of corresponding proteins to binding-sites; and k and k are self- 3 4 mathematical models with multistable dynamics. An interesting degradation rates. The model of the Z-U subsystem has the same structure question is how to embed more modules with more transcription but may have different values of model parameters. To obtain the factors to develop mathematical models with more stable steady bistability, we establish the following theorems for our proposed models for these two sub-systems. Since they have the same structure, we only states. All these issues will be interesting topics of further research. give the theorems for the X-Y sub-system. Theorem 1. There are at most ﬁve sets of non-negative equilibria for METHODS model (14). Embedding method to couple models together α k 1 3 We propose a framework to model regulatory networks with multiple 1. There are three equilibria: (0, 0), (x , 0) and (0, y ), where x ¼ e e e k β γ k 1 4 stable steady states based on the embedding of sub-systems with less and y ¼ ,if α > k and γ > k . 1 3 1 4 e k σ 4 1 B C stable steady states. It is assumed that we need to study a regulatory 2. There are two other equilibria: ðx ; y Þ and ðx ; y Þ.If > 0, > 0 1 1 2 2 A A network that consists of two regulatory modules. The ﬁrst module has and B 4AC 0, then x and x are positive real solutions of the 1 2 following equation, genes X , and it is modelled by the following equation Am þBm þC ¼ 0; (15) dX (8) ¼F ðX ; X ; ; X ; X ; ; X ; Θ ; tÞ i 1 2 n nþ1 nþN 1 dt where m ¼ β x;A¼ A B B ;B¼ A B 1 þ A B A B þ A B , 1 1 1 1 1 1 1 1 2 2 1 β γ α σ 2 1 2 1 C¼ A þ A 1 A B , A ¼ ; A ¼ ; B ¼ and B ¼ . 1 2 1 2 1 2 1 2 for i = 1, 2, ⋯ , n + N, where Θ includes model parameters of F . The σ k β k 1 i 1 3 1 4 3. To have positive values of y and y , the following conditions should 1 2 second module has the following model be satisﬁed, dY ¼G ðY ; Y ; ; Y ; Θ ; tÞ (9) j 1 2 m 2 A 1 B 1 2 2 dt x < or x < : (16) 1;2 1;2 β σ 1 2 for j = 1, 2, ⋯ , m, where Θ includes model parameters of G . In these two 2 j models, FðX; Θ ; tÞ and GðY; Θ ; tÞ are non-linear vector ﬁelds. To develop 1 2 Moreover, to study the bistability, it is necessary to establish conditions mathematical models with more stable steady states, we propose an of stability/instability for each equilibrium state. We ﬁrst give the following embedding method by assuming that X (k = 1, .. ., N) are functions of n+k conditions for each equilibrium state that locates on an axis. variables Y , Y , ⋯ , Y , given by 1 2 m Theorem 2. The X-Y system has three equilibria: (0, 0), (x , 0) and (0, y ). X ¼H ðY ; Y ; ; Y Þ: (10) e e nþk k 1 2 m 1. The equilibrium state (0, 0) is unstable if α > k and γ > k . In this way, we obtain an embedding system 1 3 1 4 2. The equilibrium state (x , 0) is stable if <k . e 4 1þσ x 2 e dW 3. The equilibrium state (0, y ) is stable if <k . e 3 1þβ y (11) ¼ FðW; Θ ; tÞ; 2 e dt In addition, we give the following stable conditions for each equilibrium where W = (X , X , ⋯ , X , Y , Y , ⋯ , Y ) represents all genes in the system, 1 2 n 1 2 m state that locates within the 2-dimensional positive real space. F denotes the embedding system from two modules with gene X and Y i i with function H . In addition, Θ = Θ ∪ Θ is the model parameters space. k 1 2 Theorem 3. The positive equilibria ðx ; y Þ and ðx ; y Þ are stable if the This embedding system (11) consists of two components: 1 1 2 2 following condition is satisﬁed. dX ¼F ðX ; X ; ; X ; H ðY ; Y ; ; Y Þ; Θ ; tÞ; i 1 2 n k 1 2 m dt β σ η ξ β σ θ ρ > 0: (17) (12) 1 2 x 1 y x 2 y dY ¼R ðX ; X ; ; X ; Y ; Y ; ; Y ; Θ ; tÞ j 1 2 n 1 2 m dt where θ = 1 + β x, η = 1 + β y, ρ = 1 + σ y and ξ = 1 + σ x. x 1 y 2 y 1 x 2 for i = 1, 2, ⋯ , n, k = 1, .. ., N and j = 1, 2, ⋯ , m. Since each X is regulated by the X (k = 1, .. ., N), and X are functions of Y , Y , ⋯ , Y , the In summary, Theorem 1 gives the existence conditions of the equilibria n+k n+k 1 2 m expressions of each gene Y is also regulated by X (i = 1, .. ., n). The non- for our proposed two-node systems. Theorems 2 and 3 provide the j i linear vector ﬁeld GðY; Θ ; tÞ in Eq. (9) will then be transformed into a new necessary conditions for stability properties of these equilibria. According non-linear vector ﬁeld RðW; Θ ; tÞ, which includes both genes X and Y to these theorems, we can easily check whether two-node systems have i i from two sub-systems with their corresponding regulations. Note that this bistability based on generated samples of model parameters. The proofs of is a general idea to develop mathematical models with more stable steady these theorems are given in Supplementary Notes. states. Depending on the speciﬁc formalism and properties of sub-systems, the embedding system may have different results regarding multiple Perturbation analysis of bistable models stable steady states with different conditions. In this study, we only focus We have proved that systems (13)and (14) have bistable steady states on the systems with Shea-Ackers formalism . under the conditions in Theorems 2 or 3. Next we use the random search method to ﬁnd the model parameters with which the system has bistable Model development for bistability properties steady states. We ﬁrst generate a sample for each model parameter from We ﬁrst develop a model for the network in Fig. 1c with bistability the uniform distribution over the interval [0, A] and then test whether the properties. Suppose that two sub-systems, namely the Z-U system and X-Y system with the sampled parameters satisﬁes the conditions in Theorems sub-system, have the same structure of a double-negative feedback loop 2 or 3. If the conditions are satisﬁed, we solve nonlinear equations of the and positive autoregulations. For the Z-U system, based on the formalism system to ﬁnd the steady states. We test different values of A and ﬁnd (8) with X = {z, u} and Θ = {a , b , b , c , d , d , k , k }, we propose the that the system has bistable steady states when A = 10. To ﬁnd more 1 1 1 2 1 1 2 1 2 Published in partnership with the Systems Biology Institute npj Systems Biology and Applications (2022) 10 S. Wu et al. types of bistable states, we test 10000 sets of parameters from the DATA AVAILABILITY uniform distribution over the interval [0, 10]. Supplementary Table 1 gives No datasets were generated during the current study. The experimental data for three types of bistable steady states, namely Case 1: (x , 0) and (0, y ); e e erythroiesis and granulopoiesis that support the parameter estimation of this study Case 2: (x ,0) and ðx ; y Þ; and Case 3: (0, y )and ðx ; y Þ: All stable states e e are available in the published paper at https://www.nature.com/articles/s41586-020- 1 1 2 2 in case 1 are located on the coordinate axis. We add a perturbation to 2432-4 . each estimated coefﬁcient c as c = [ε ×(P − 0.5) + 1] × c,where P is a uniformly distributed random variable over the interval [0, 1], and ε is the strength of perturbation. Supplementary Table 2 shows that the two CODE AVAILABILITY other cases of bistability can be obtained by the perturbed coefﬁcients The code used to perform the analyses presented in the current study is available from Case 1. from the corresponding author on reasonable request. Model development for tristability properties Received: 14 July 2021; Accepted: 15 February 2022; The 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 ¼Hðx; 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 REFERENCES expressions of genes x and y are also negatively regulated by gene z in the new embedding model. The non-linear vector ﬁelds G ðx; y; Θ ; tÞ 1;2 1 1. Ozbudak, E. M., Thattai, M., Lim, H. N., Shraiman, B. I. & Oudenaarden, A. V. are then transformed into new non-linear vector ﬁelds R ðx; y; z; Θ ; tÞ, 1;2 Multistability in the lactose utilization network of Escherichia coli. Nature 427, respectively, which include genes x, y and z from two sub-systems with 737–740 (2004). negative regulations from gene z to genes x and y. Using the embedding 2. Li, Q. et al. 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A robust method for designing multistable systems by embedding bistable subsystems

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A robust method for designing multistable systems by embedding bistable subsystems

A robust method for designing multistable systems by embedding bistable subsystems

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