A robust method for designing multistable systems by embedding bistable subsystems

Abstract

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.

Full text

ARTICLE OPEN
A robust method for designing multistable systems by
embedding bistable subsystems
Siyuan Wu
1
, Tianshou Zhou
2
and Tianhai Tian
1
✉
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
Multistability is the characteristic of a system that exhibits two or
more mutually exclusive stable states. This phenomenon has been
observed in many different disciplines of science, including
genetic regulatory networks
1–4
, cell signalling pathways
5–8
, meta-
bolic networks
9
, ecosystems
10,11
, neuroscience
12
, laser sys-
tems
13,14
, and quantum systems
15
. When external and/or
internal conditions change, the system may switch from one
steady state to another either randomly by perturbations or in a
desired way according to the control strategies. In recent years
mathematical models with multistability have been developed for
theoretical analysis and computer simulations, which shed light on
the mechanisms that generate multistability and control the
transition between steady states
16–19
.
As one of the important molecular systems showing multi-
stability, hematopoiesis is a highly integrated developmental
process that controls the proliferation, differentiation and
maturation of hematopoietic stem cells (HSCs)
20,21
. HSCs have
the features of self-renewal and multipotency as well as the ability
to differentiate into multipotent progenitors (MPPs). Each of these
cell types is regarded as a stable state of the multistable system. In
addition, the formation of white and red blood cells is a dynamical
process that transits a cell from one stable cell type to another.
This process begins with the differentiation of HSCs and enters the
main stage at which cells reach either common myeloid
progenitors (CMPs) or common lymphoid progenitors (CLPs)
22,23
.
Transcription factors play a key role in controlling the process of
blood cell lineage specification. Experimental studies have
demonstrated that the genetic module GATA1-PU.1 is a vital
component for the fate commitment of CMPs between erythro-
poiesis and granulopoiesis
24,25
. HSCs are more likely to choose
megakaryocyte/erythroid progenitors (MEPs) with high expression
levels of GATA1
26
, or conversely to choose granulocyte/macro-
phage progenitors (GMPs) with high expression levels of PU.1
27
.In
addition, the regulation between genes GATA1 and GATA2 is an
essential driver of hematopoiesis
28
. Experimental studies
suggested that GATA2 and GATA1 sequentially bind the same
cis-elements, which is referred to as the GATA-switching
29,30
.
Mathematical modelling is a powerful tool to accurately
describe the dynamics of hematopoiesis and to explore the
regulatory mechanisms for controlling the transitions between
different cell types
31–37
. For the GATA1-PU.1 module, Hill equations
with high cooperativity were initially used to realise tristability
38
.In
addition, mathematical models have been proposed to achieve
bistability in gene regulatory networks without any high
cooperativity coefficients
39,40
. Bifurcation theory is also an efficient
method to explore the mechanisms of GATA1-PU.1 module
41
.We
have proposed a mathematical model to realise the mechanisms
of GATA-switching and designed an effective algorithm to realise
tristability of mathematical models
42
. Moreover, the underlying
mechanisms of how the stem/progenitor cells leave the stable
steady states and commit to a specific lineage were also revealed
with the assistance of mathematical models
43
. At the single cell
level, the differentiation processes of embryonic stem cells were
simulated by Langevin equations, which helped to identify
potential transcriptional regulators of lineage decision and
commitment
44
. Mathematical models have also been used to
study the dynamical properties of diseases such as periodic
haematological disorders
45
.
Although these attempts have realised tristability by using
different assumptions, it is still a challenge to develop mathema-
tical models to realise tristability using both the realistic regulatory
mechanisms and experimental data. On the other hand,
substantial research studies have been conducted to develop
mathematical models for realising bistability properties
3,46–51
.
Thus, the question is whether we can develop mathematical
models with tristability or higher order of multistability by using
bistable models. To address this issue, we propose a robust
method to develop multistable models by embedding bistable
models together. Using the GATA1-GATA2-PU.1 module as a
testing model, we develop a tristable model based on two
systems that have no high cooperativity coefficients.
1
School of Mathematics, Monash University, Melbourne, VIC, Australia.
2
School of Mathematics and Statistics, Sun Yet-Sen University, Guangzhou, China.
✉email: Tianhai.Tian@monash.edu
www.nature.com/npjsba
Published in partnership with the Systems Biology Institute
1234567890():,;

RESULTS
Embedding method for designing multistable models
The 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 Uin the first Z-U
module is an auxiliary node, which is assumed to be U=μX+δY,
where μand δare two positive parameters. When the system stays
in the state with a high expression level of Zand a low level of U,
the expression levels of Xand Yare low. However, when the
system has a low expression level of Zand a high level of U, the
system triggers the second module X-Yto choose either a high
level of Xand a low level of Yor a low level of Xand a high level of
Y. In this way we realise the system with three stable states in
which one of the three variables (namely Z,Xor Y) is at the high
expression state but the other two are at low expression states.
To demonstrate the effectiveness of the proposed embedding
method, we use the toggle switch network as the test system
52
.
This network consists of two genes that form a double negative
feedback loop and is modelled by the following equations with
parameter space Θ
1
={a=0.2, b=4, c=3}, given by
dz
dt ¼F
1ðz;u;Θ1;tÞ¼0:2þ4
1þu3z;
du
dt ¼F
2ðz;u;Θ1;tÞ¼0:2þ4
1þz3u:
(1)
It is assumed that the first Z-Umodule follows model (1) and the
second X-Ymodule satisfies the same model with same parameter
space Θ
1
, but different variables xand y, given by
dx
dt ¼G
1ðx;y;Θ1;tÞ¼0:2þ4
1þy3x;
dy
dt ¼G
2ðx;y;Θ1;tÞ¼0:2þ4
1þx3y:
(2)
Now we embed these two sub-systems together using
u¼Hðx;yÞ¼xþy. Since gene zis negatively regulated by gene
uin the sub-system (1), and uis a function of genes xand y, the
expressions of genes xand yare also negatively regulated by gene
zin the new embedding model. Then the non-linear vector fields
G1;2ðx;y;Θ1;tÞare transformed into new non-linear vector fields
R1;2ðx;y;z;Θ1;tÞ, respectively, which include genes x,yand z
from two sub-systems with negative regulations from gene zto
genes xand y. Therefore, the new model with three variables is
given by
dx
dt ¼R
1ðx;y;z;Θ1;tÞ¼0:2þ4
ð1þy3Þð1þz3Þx;
dy
dt ¼R
wðx;y;z;Θ1;tÞ¼0:2þ4
ð1þx3Þð1þz3Þy;
dz
dt ¼F
1ðz;u¼xþy;Θ1;tÞ¼0:2þ4
1þðxþyÞ3z:
(3)
Figure 2a shows the phase plane of the toggle switch sub-system
(1) with bistability properties, and Fig. 2b provides the 3D phase
portrait of the embedded model (3) with three stable steady
states. The embedded model successfully realised the tristability,
which validates our embedding method for developing mathe-
matical models with multistability.
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. bThe principle of embeddedness: Z-Umodule is the first
bistable sub-system. Once this module crosses the saddle point from state Zto state U, it enters the X-Ysub-system that has two stable steady
states Xand Y, reaching either state Xor state Yvia the auxiliary state U.c,dThe structure of two double-negative feedback loops with positive
autoregulations, which is the mechanisms for bistable sub-systems in HSCs. eThe structure and mathematical model of regulatory network
after embeddedness. The X-Ysub-system is embedded into the state U.
S. Wu et al.
2
npj Systems Biology and Applications (2022) 10 Published in partnership with the Systems Biology Institute
1234567890():,;

Bistable 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-Umodule (13) and X-Ymodule (14). These two
models have the same structure but with different model
parameters. Theorem 1 shows that there are five possible non-
negative equilibria in these models. Theorem 2 indicates that two
steady states located on the axis are stable under the given
conditions. In addition, Theorem 3 gives the conditions under
which two possible steady states located out of the axis are stable
(see Methods).
We further search for stable steady states of the model with
randomly sampled parameters. Supplementary Table 1 gives
three types of bistable steady states. However, we have not
found any parameter samples to realise tristability. To test
robustness properties, we conduct perturbation tests by
examining the bistable property of the model with slightly
changed model parameters
53,54
. Our computational results
demonstrate that a perturbed bistable model with one stable
steady state located on the axis but another located off the axis
canbefoundforamodelwithtwostable steady states located
ontheaxis(seeSupplementaryTable2).Theseresultssuggest
that the developed model has very good robustness properties
in terms of parameter variations.
We next use the approximate Bayesian computation (ABC)
rejection algorithm
55,56
to estimate model parameters based on
the experimental data for erythroiesis and granulopoiesis
21
.We
first estimate parameters in the X-Ymodule that describes
regulations between genes GATA1 and PU.1(14). It is assumed
that the prior distribution of each parameter is a uniform
distribution over the interval [0, 100]. The distance between
experimental data and simulations is measured by
ρðX;XÞ¼X
m
i¼1
½jxix
ijþjyiy
ij;
where (x
i
,y
i
) and (x
i;y
i) are the observed data and simulated data
for genes (X,Y), respectively. Supplementary Table 3 gives the
estimated parameters of this module. Figure 3a shows that the
phase plane of the GATA1-PU.1 sub-system based on estimated
parameters, which shows that this system is bistable.
Regarding the Z-Umodule (13) that describes the regulation of
GATA-switching, to be consistent with the module structure, we
first assume that GATA1 and GATA2 form a double negative
feedback module with autoregulations, and will modify this
assumption later based on the experimentally observed mechan-
ism. Here the data of the auxiliary variable Uis the sum of GATA1
and PU.1. Supplementary Table 4 gives the estimated parameters
of the Z-Umodule.
An experimental study has identified GATA2 at chromatin sites
in early-stage erythroblasts
28
, when expression levels of GATA1
increase as erythropoiesis progresses, GATA1 displaces GATA2
from chromatin sites. To describe the mechanism of GATA-
switching, we introduce an additional rate constant k
*
over a time
interval [t
1
,t
2
] for the displacement rate of GATA2 proteins during
the process of GATA-switching, given by
k¼k
0t2½t1;t2;
0 otherwise :
(4)
Since the displacement of GATA2 protein increasing, the
concentration of GATA1 proteins around the binding site will
increase proportionally to k
*
. Hence, we use rate ψk
*
zfor the
increase of GATA1 during GATA-switching, where ψis a control
parameter to adjust the availability of GATA1 proteins around
chromatin sites. Then the GATA-switching module is modelled by
dz
dt ¼a1z
1þb1z
1
1þb2uk1zkz;
du
dt ¼c1u
1þd1u
1
1þd2zk2uþψkz;
(5)
where zand uare expression levels of GATA2 and GATA1,
respectively. Note that the bistability property of this module is
realised by model (5) using k
*
=0. Figure 3b gives two simulations
for an unsuccessful switching and a successful switching. It is
assumed that the GATA-switching occurs over the interval [t
1
,t
2
]
=[500, 3500]. Simulations show that an adequate displacement of
GATA2 is the key to achieve GATA-switching using a relatively
large value of k
01.
Tristable model of the GATA1-GATA2-PU.1 network
After successfully realising the bistability in double-negative
feedback loops with positive autoregulation, we next incorpo-
rate the GATA1-PU.1 regulatory module into the GATA-
switching module to realise the tristability of HSC differentia-
tion. We use expression levels of GATA1 in the GATA-switching
module to represent total levels of GATA1 plus PU.1,andembed
these two modules together (18)(seeTheorems4–6inMethods
for more details). The model parameters have the same values
as the corresponding parameters in the Z-Umodule 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). bThe 3D phase portrait of the embedded system (3) with tristability (Three red
points: stable steady states; two black points: saddle states).
S. Wu et al.
3
Published in partnership with the Systems Biology Institute npj Systems Biology and Applications (2022) 10

the embedded system, which shows that the embedding
model faithfully realises three stable steady states, which also
suggests that the proposed embedding method is a robust
approach to develop high order multistable models based on
bistable models.
As mentioned in the previous subsection, the GATA-switching
module is not a perfect double-negative feedback loop. In fact,
experimental studies suggest that GATA2 moderately simulates
the expression of gene GATA1
57
. Thus we make a modification to
model (18) by adding the term d
*
zin the first equation to
represent a weak positive regulation from GATA2 to GATA1.In
addition, to avoid zero basal gene expression levels, we add a
constant to each equation of the proposed model (18). The
modified model is given by,
dx
dt ¼α0þα1x
1þβ1x
1
1þβ2y
1þdz
1þd2zk3xþψkz;
dy
dt ¼γ0þγ1y
1þσ1y
1
1þσ2x
1
1þd2zk4y;
dz
dt ¼a0þa1z
1þb1z
1
1þb2ðxþyÞk1zkz;
(6)
where x,y,zrepresent expression levels of genes GATA1,GATA2
and PU.1, respectively. The values of α
0
,γ
0
,a
0
and d
*
are carefully
selected so that the model simulation still matches experimental
data and the model has at least three stable steady states (see
Supplementary Table 5). Figure 3c gives the 3D phase portrait of
system (6) with k
*
=0. Using estimated parameters (see Supple-
mentary Tables 3–5), the modified system (6) actually achieves
quad-stability. In three stable states, one of the three genes has
high expression levels but the other two have low expression
levels. The fourth stable state has low expression levels (2.3364,
0.7417, 8.6664) of the three genes. In fact, these are exact four
transcriptional states that have been observed in experimental
studies, namely a PU.1
high
Gata1/2
low
state (P1H); a Gata1
high-
GATA2/PU.1
low
state (G1H); a Gata2
high
GATA1/PU.1
low
state (G2H);
and a state with low expression of all three genes (LES CMP)
21
.
Compared with existing modelling studies, our embedding model
(6) successfully realises the state with low expression levels of all
three genes.
Note that the embedding model is based on the assumption of
GATA-switching, namely the exchange of GATA1 for GATA2 at the
chromatin site, which controls the expression of genes GATA1 and
GATA2. However, a low level of GATA2 at the chromatin site does
not mean the total level of GATA2 in cells is also low. This may be
the reason for the difference between the simulated state
Gata1
high
GATA2/PU.1
low
state (G1H) (namely only GATA1 has high
expression) and the experimentally observed state Gata1/2
high-
PU.1
low
state (G1/2H) (namely both GATA1 and GATA2 have high
expression levels)
21
.
Stochastic model for realising heterogeneity
Although the modified embedding model has successfully realised
the quad-stability properties, this deterministic model cannot
describe the heterogeneity in the cell fate commitment. Thus, the
next question is whether we can use a stochastic model to realise
experimental data showing different gene expression levels in
single cells
21
. To answer this question, we propose a stochastic
differential equations model in Itô form to describe the functions
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. bSimulations of
GATA-switching of model (5). Upper panel: An unsuccessful switching with a small value of k
0due to the displacement of GATA2 not being
enough for cells to leave the HSCs state (Zstate); Lower panel: A successful switching with sufficient displacement of GATA2 by using a large
value of k
0. Cells leave the HSCs state and enter the Ustate. cThe 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.
S. Wu et al.
4
npj Systems Biology and Applications (2022) 10 Published in partnership with the Systems Biology Institute

of noise during the cell lineage specification, given by (7)
dXðtÞ¼ α0þα1XðtÞ
1þβ1XðtÞ
1
1þβ2YðtÞ
1þdZðtÞ
1þd2ZðtÞk3XðtÞþψkZðtÞ
hi
dt þ½ω1ðk3XðtÞþψkZðtÞÞdW1
t;
dYðtÞ¼ γ0þγ1YðtÞ
1þσ1YðtÞ
1
1þσ2XðtÞ
1
1þd2ZðtÞk4YðtÞ
hi
dt þ½ω2k4YðtÞdW2
t;
dZðtÞ¼ a0þa1ZðtÞ
1þb1ZðtÞ
1
1þb2ðXðtÞþYðtÞÞ k1ZðtÞkZðtÞ
hi
dt þ½ω3ðk1þkÞZðtÞdW3
t;
(7)
where W1
t,W2
tand W3
tare three independent Wiener processes
whose increment is a Gaussian random variable ΔW
t
=W(t+Δt)
−W(t)~N(0, Δt), and ω
1
,ω
2
and ω
3
represent noise strengths. The
reason for selecting Itô form is to maintain the mean of the
stochastic system (7) as the corresponding deterministic system
(6). To test the influence of GATA-switching on determining the
transitions between different states, we introduce noise to
coefficient k
*
and consequently to the three degradation
processes in the model. We use the semi-implicit Euler method
to simulate the proposed model
58
. Figure 4provides four
stochastic simulations for four different types of cell fate
commitments with model parameters k
0¼0:52, ψ=0.0005, ω
1
=0.04, and ω
2
=ω
3
=0.08. Figure 4a, b shows two simulations of
unsuccessful GATA switching when the displacement of GATA2 is
not sufficient. However, a sufficient displacement of GATA2 can
trigger successful GATA switching, which leads to either the GMP
state with high expression levels of PU.1 in Fig. 4c or the MEP state
with high expression levels of GATA1 in Fig. 4d.
To examine the heterogeneity of hematopoiesis with different
displacement rates k
0and ψtogether, we generate 20,000 sto-
chastic simulations for each set of k
0and ψvalues over the range
of [0.04, 1] and [0, 0.001], respectively. The ranges of k
0and ψare
determined by numerical testing. If all stochastic simulations
move to a single stable state for the given k
0and ψvalues, we
change the lower bound and/or upper bound of the value range
in order that simulations may move to different stable states for
the given k
0and ψvalues. To show the boundary of parameter
space, we also keep certain sets of parameter values with which
simulations move to one specific stable state. Figure 5a gives
proportions of simulations that have successful switching in
20,000 simulations. When the value of k
0is between 0.1 and 0.2,
the displacement speed of GATA2 is low, which gives limited relief
of negative regulation to PU.1, but GATA1 increases gradually due
to GATA-switching and weak positive regulation from GATA2 to
GATA1. Thus nearly all cells choose the MEP state with high
expression levels of GATA1. However, if the value of k
0is larger,
the negative regulation from GATA2 to PU.1 is eliminated quickly,
thus the competition between GATA1 and PU.1 will lead cells to
different lineages. When the value of k
0is relatively large but the
value of ψis relatively small, the increase of GATA1 is slow due to
the smaller value of ψin GATA-switching. However, the negative
regulation from GATA2 to PU.1 declines rapidly due to the larger
value of k
0. Thus, Fig. 5b shows that the combination of larger k
0
and smaller ψvalues allows more cells to move to the GMP
lineage with high expression level of PU.1. If there is no winner in
the competition between GATA1 and PU.1, the cell then moves to
the state with low expression levels of three genes (namely LE3G).
Figure 5c shows that, when the value of k
0is larger than 0.2, there
are four types of simulations as shown in Fig. 5for a set of k
0and
ψvalues. We use a MATLAB package
59
to give the violin plot for
the expression distributions of three genes in three different
cellular states. The violin plot is a combination of a box plot and a
kernel density plot that illustrates data peaks. The violin plots in
Fig. 5d match the experimental observations very well
21
.
Regarding the size of basins of attraction, we first calculate the
distances between the stable states and saddle points in Fig. 3c,
which are given in Supplementary Table 6. The minimal distance
between the G1H state and three saddle points is much larger
than the minimal distances of the other three stable states to the
Fig. 4 Stochastic simulations showing four stable states that correspond to the experimentally observed four different states.
aSimulation of unsuccessful GATA switching that makes the cell stay at the HSC state, which is the G2H state. bSimulation of unsuccessful
GATA switching but the cell enters the state with low expression of all three genes, which is the LES CMP state. cSimulation of successful
switching that leads to the GMP state with high expression levels of PU.1, which is the P1H state. dSimulation of successful switching that
leads to the MEP state with high expression levels of GATA1, which is the G1H state.
S. Wu et al.
5
Published in partnership with the Systems Biology Institute npj Systems Biology and Applications (2022) 10

saddle points, which suggests that the size of basin of attraction
for the G1H state is larger than those of the other three stable
states. In addition, we observe the variability of stable states in
20,000 stochastic simulations. Supplementary Table 7 shows that
the variations of GATA1 in the G1H state are much larger than
those of the other two genes when having high expression levels.
We also study the relative frequency of LE3G state. Supple-
mentary Fig. 2 shows that, for a fixed value of parameter ψ, the
frequency increases as the value of k
0increases. In addition, for a
fixed value of k
0, the frequency decreases as the value of ψ
increases. The variation of parameter ψis much more important
than that of parameter k
0. For the simulations showing in Fig. 5d,
the frequency is 0.1080 with k
0¼0:52 and ψ=0.0005. Figure 5d
and Supplementary Fig. 2 suggest that more cells remain in the
LE3G or P1H (GMP) state if GATA2 leaves the chromatin site fast
(i.e. a large k
0value) and the expression of GATA1 is slow (i.e. a
small ψvalue). However, if the expression of GATA1 is fast (i.e. a
large ψvalue), more cells will transit to G1H (MEP) state and the
frequency of the LE3G state is low, which is consistent with the
results in a recent study
60
.
DISCUSSIONS
Inspired by Waddington’s epigenetic landscape model, we
assume that a multistable system makes a series of binary
decisions for the selection of multiple evolutionary pathways.
Compared with modelling studies for multistable networks, it is
relatively easy to develop models with bistability and there is a
rich literature for studying bistable networks. Thus, our proposed
embedding method is an effective approach to develop multi-
stable models based on well-studied models with bistable
properties. In addition, using cell fate commitments in hemato-
poiesis as the test problem, we have successfully realised
tristability in the GATA-PU.1 module by embedding two bistable
modules together. More importantly, by modifying the model
using experimentally determined regulatory mechanisms, the
developed model successfully realises four stable states that have
been observed in a recent experimental study
21
.
In this study the stable states are achieved by a model without
high cooperativity (i.e. Hill coefficient n=1). Recently, the
dynamics of toggle triad with self-activations have attracted
much attention
60,61
. Mathematical models with high cooperativity
have been developed to achieve pentastable, namely a hybrid X/Y
state with high X, high Yand low Z. We tried to realise
pentastability by using our proposed model with high coopera-
tivity (n=2 or 3), but numerical tests were not successful. Thus,
high cooperativity in self-activation may be essential to realise
pentastable. This is an interesting problem that will be the topic of
further studies.
Despite 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,thestateUis 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 levels or high PU.1
levels or return to the stem cell state. These simulations are
consistent with the CLOUD-HSPC model in which differentiation
Fig. 5 Distributions of different cell types derived from stochastic simulations. a Frequencies of cells having successful switching for each
set of parameters ðk
0;ψÞ.bRatios of GMP cells to MEP cells when cells have successfully switched in afor each set of parameters ðk
0;ψÞ.
cParameter sets of ðk
0;ψÞthat generate stochastic simulations with four steady states as shown in Fig. 4(yellow part) or with two or three
states (blue part). dViolin 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
0¼0:52 and ψ=0.0005.
S. Wu et al.
6
npj Systems Biology and Applications (2022) 10 Published in partnership with the Systems Biology Institute

is a process of uncommitted cells in transitory states that
gradually acquire uni-lineage priming
62–64
. In addition, stochastic
simulations demonstrate that noise plays a key role in
determining different differentiation pathways.
This work uses differential equation models to determine stable
steady states and then employs corresponding stochastic models
to realise the functions of noise. However, experimental studies
have shown that gene expression is a bursting process. The
challenge is how to determine conditions for realising the
multistable properties in stochastic models with bursting pro-
cesses. In addition, hematopoiesis is a process to produce all
mature blood cells. This is an ideal test system to develop
mathematical models with multistable dynamics. An interesting
question is how to embed more modules with more transcription
factors to develop mathematical models with more stable steady
states. All these issues will be interesting topics of further research.
METHODS
Embedding method to couple models together
We propose a framework to model regulatory networks with multiple
stable steady states based on the embedding of sub-systems with less
stable steady states. It is assumed that we need to study a regulatory
network that consists of two regulatory modules. The first module has
genes X
i
, and it is modelled by the following equation
dXi
dt ¼F
iðX1;X2; ;Xn;Xnþ1; ;XnþN;Θ1;tÞ(8)
for i=1, 2, ⋯,n+N, where Θ
1
includes model parameters of Fi. The
second module has the following model
dYj
dt ¼G
jðY1;Y2; ;Ym;Θ2;tÞ(9)
for j=1, 2, ⋯,m, where Θ
2
includes model parameters of Gj. In these two
models, FðX;Θ1;tÞand GðY;Θ2;tÞare non-linear vector fields. To develop
mathematical models with more stable steady states, we propose an
embedding method by assuming that X
n+k
(k=1,...,N) are functions of
variables Y
1
,Y
2
,⋯,Y
m
, given by
Xnþk¼H
kðY1;Y2; ;YmÞ:(10)
In this way, we obtain an embedding system
dW
dt ¼FðW;Θ
;tÞ;(11)
where W=(X
1
,X
2
,⋯,X
n
,Y
1
,Y
2
,⋯,Y
m
) represents all genes in the system,
Fdenotes the embedding system from two modules with gene X
i
and Y
i
with function Hk. In addition, Θ
*
=Θ
1
∪Θ
2
is the model parameters space.
This embedding system (11) consists of two components:
dXi
dt ¼F
iðX1;X2; ;Xn;HkðY1;Y2; ;YmÞ;Θ
;tÞ;
dYj
dt ¼R
jðX1;X2; ;Xn;Y1;Y2; ;Ym;Θ
;tÞ(12)
for i=1, 2, ⋯,n,k=1,...,Nand j=1, 2, ⋯,m. Since each X
i
is regulated
by the X
n+k
(k=1,...,N), and X
n+k
are functions of Y
1
,Y
2
,⋯,Y
m
, the
expressions of each gene Y
j
is also regulated by X
i
(i=1,...,n). The non-
linear vector field GðY;Θ2;tÞin Eq. (9) will then be transformed into a new
non-linear vector field RðW;Θ
;tÞ, which includes both genes X
i
and Y
i
from two sub-systems with their corresponding regulations. Note that this
is a general idea to develop mathematical models with more stable steady
states. Depending on the specific formalism and properties of sub-systems,
the embedding system may have different results regarding multiple
stable steady states with different conditions. In this study, we only focus
on the systems with Shea-Ackers formalism
65
.
Model development for bistability properties
We first develop a model for the network in Fig. 1c with bistability
properties. Suppose that two sub-systems, namely the Z-Usystem and X-Y
sub-system, have the same structure of a double-negative feedback loop
and positive autoregulations. For the Z-Usystem, based on the formalism
(8) with X={z,u} and Θ
1
={a
1
,b
1
,b
2
,c
1
,d
1
,d
2
,k
1
,k
2
}, we propose the
following model to describe the dynamics, given by
dz
dt ¼F
1ðz;u;Θ1;tÞ¼ a1z
1þb1z
1
1þb2uk1z;
du
dt ¼F
2ðz;u;Θ1;tÞ¼ c1u
1þd1u
1
1þd2zk2u:
(13)
Similarly, based on the formalism (9) with Y={x,y} and Θ
2
={α
1
,β
1
,β
2
,γ
1
,
σ
1
,σ
2
,k
3
,k
4
}, the dynamics of the X−Ysubsystem is modelled by
dx
dt ¼G
1ðx;y;Θ2;tÞ¼ α1x
1þβ1x
1
1þβ2yk3x;
dy
dt ¼G
2ðx;y;Θ2;tÞ¼ γ1y
1þσ1y
1
1þσ2xk4y;
(14)
where xand yare expression levels of genes Xand Y, respectively; α
1
and
γ
1
represent expression rates; β
1
,β
2
,σ
1
and σ
2
represent association rates
of corresponding proteins to binding-sites; and k
3
and k
4
are self-
degradation rates. The model of the Z-Usubsystem has the same structure
but may have different values of model parameters. To obtain the
bistability, we establish the following theorems for our proposed models
for these two sub-systems. Since they have the same structure, we only
give the theorems for the X-Ysub-system.
Theorem 1. There are at most five sets of non-negative equilibria for
model (14).
1. There are three equilibria: (0, 0), (x
e
, 0) and (0, y
e
), where xe¼α1k3
k3β1
and ye¼γ1k4
k4σ1,ifα
1
>k
3
and γ
1
>k
4
.
2. There are two other equilibria: ðx
1;y
1Þand ðx
2;y
2Þ.IfB
A>0, C
A>0
and B24AC  0, then x
1and x
2are positive real solutions of the
following equation,
Am2þBmþC¼ 0;(15)
where m¼β1x;A¼A1B1B1;B¼A1B11þA1B1A1B2þA2B1,
C¼A1þA21A1B2,A1¼β2
σ1;A2¼α1
k3;B1¼σ2
β1and B2¼γ1
k4.
3. To have positive values of y
1and y
2, the following conditions should
be satisfied,
x
1;2<A21
β1
or x
1;2<B21
σ2
:(16)
Moreover, to study the bistability, it is necessary to establish conditions
of stability/instability for each equilibrium state. We first give the following
conditions for each equilibrium state that locates on an axis.
Theorem 2. The X-Ysystem has three equilibria: (0, 0), (x
e
, 0) and (0, y
e
).
1. The equilibrium state (0, 0) is unstable if α
1
>k
3
and γ
1
>k
4
.
2. The equilibrium state (x
e
, 0) is stable if γ1
1þσ2xe<k
4.
3. The equilibrium state (0, y
e
) is stable if α1
1þβ2ye<k
3.
In addition, we give the following stable conditions for each equilibrium
state that locates within the 2-dimensional positive real space.
Theorem 3. The positive equilibria ðx
1;y
1Þand ðx
2;y
2Þare stable if the
following condition is satisfied.
β1σ1ηyξxβ2σ2θxρy>0:(17)
where θ
x
=1+β
1
x,η
y
=1+β
2
y,ρ
y
=1+σ
1
yand ξ
x
=1+σ
2
x.
In summary, Theorem 1 gives the existence conditions of the equilibria
for our proposed two-node systems. Theorems 2 and 3 provide the
necessary conditions for stability properties of these equilibria. According
to these theorems, we can easily check whether two-node systems have
bistability based on generated samples of model parameters. The proofs of
these theorems are given in Supplementary Notes.
Perturbation analysis of bistable models
We have proved that systems (13)and(14) have bistable steady states
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
steady states. We first generate a sample for each model parameter from
the uniform distribution over the interval [0, A] and then test whether the
system with the sampled parameters satisfies the conditions in Theorems
2 or 3. If the conditions are satisfied, we solve nonlinear equations of the
system to find the steady states. We test different values of Aand find
that the system has bistable steady states when A=10. To find more
S. Wu et al.
7
Published in partnership with the Systems Biology Institute npj Systems Biology and Applications (2022) 10

types of bistable states, we test 10000 sets of parameters from the
uniform distribution over the interval [0, 10]. Supplementary Table 1 gives
three types of bistable steady states, namely Case 1: (x
e
, 0) and (0, y
e
);
Case 2: (x
e
,0)and ðx
1;y
1Þ; and Case 3: (0, y
e
)andðx
2;y
2Þ:All stable states
in case 1 are located on the coordinate axis. We add a perturbation to
each estimated coefficient cas c
*
=[ε×(P−0.5) +1] × c,wherePis a
uniformly distributed random variable over the interval [0, 1], and εis the
strength of perturbation. Supplementary Table 2 shows that the two
other cases of bistability can be obtained by the perturbed coefficients
from Case 1.
Model development for tristability properties
The mathematical model for the network of three genes is formed by
embedding the X-Ysystem into the Z-Usystem as shown in Fig. 1d. For
simplicity, let u¼Hðx;yÞ=x+y.Sincegenezis negatively regulated by
gene uin sub-system (13), and uis a function of genes xand y,the
expressions of genes xand yare also negatively regulated by gene zin
the new embedding model. The non-linear vector fields G1;2ðx;y;Θ1;tÞ
are then transformed into new non-linear vector fields R1;2ðx;y;z;Θ
;tÞ,
respectively, which include genes x,yand zfrom two sub-systems with
negative regulations from gene zto genes xand y. Using the embedding
method (12) and sub-system models ((13), (14)), we obtain the following
model to describe the embedded X-Y-Zsystem,
dx
dt ¼R
1ðx;y;z;Θ
;tÞ¼ α1x
1þβ1x
1
1þβ2y
1
1þd2zk3x;
dy
dt ¼R
2ðx;y;z;Θ
;tÞ¼ γ1y
1þσ1y
1
1þσ2x
1
1þd2zk4y;
dz
dt ¼F
1ðz;u¼xþy;Θ
;tÞ¼ a1z
1þb1z
1
1þb2ðxþyÞk1z:
(18)
To verify the tristability of model (18), we give the following conditions for
existence of the equilibria and necessary conditions for stability properties
of these equilibria.
Theorem 4.
1. If (x
e
, 0) and (0, y
e
) are equilibria of X-Ysub-system and (z
e
,0) is a
equilibrium state of Z-Usub-system, where xe¼α1k3
k3β1,ye¼γ1k4
k4σ1and
ze¼a1k1
k1b1, then (x
e
, 0, 0), (0, y
e
, 0) and (0, 0, z
e
) are three equilibria of
the embedding system (18).
2. If ðx
1;y
1Þand ðx
2;y
2Þare two positive equilibria of X-Ysystem as
stated in Theorem 1, then ðx
1;y
1;0Þand ðx
2;y
2;0Þare still two
equilibria of the embedding system (18).
This theorem shows that existence conditions of equilibria in the
embedded system are the same as those of two-node sub-systems. Thus,
the information of two-node sub-systems can be directly applied to the
embedded system. For each equilibrium state located on the axis, we give
the following conditions of stability.
Theorem 5.If(x
e
, 0) and (0, y
e
) are both stable states of X-Ysystem and (z
e
,0)
is a stable state of Z-Usystem.
1. The equilibrium state (x
e
, 0, 0) is stable if a1
1þb2xe<k
1.
2. The equilibrium state (0, y
e
, 0) is stable if a1
1þb2ye<k
1.
3. The equilibrium state (0, 0, z
e
) is stable if α1
1þd2ze<k3and γ1
1þd2ze<k
4.
In addition, we give the following stable conditions for each equilibrium
state that locates within the 3-dimensional positive real space.
Theorem 6. Suppose (x
*
,y
*
) is a stable state of X-Ysystem, then the
equilibrium state (x
*
,y
*
, 0) is also a stable state of the X-Y-Zsystem if
a1
1þb2ðxþyÞ<k1:(19)
Theorems 5 and 6 describe the necessary conditions for stability
properties of the equilibria in the embedding X-Y-Zsystem. By applying
these theorems, we can further constrain the estimated parameters
obtained from two-node systems so that the embedding system can
achieve tristability. The proofs of Theorems 4–6 are given in Supplemen-
tary Notes.
DATA AVAILABILITY
No datasets were generated during the current study. The experimental data for
erythroiesis and granulopoiesis that support the parameter estimation of this study
are available in the published paper at https://www.nature.com/articles/s41586-020-
2432-4
21
.
CODE AVAILABILITY
The code used to perform the analyses presented in the current study is available
from the corresponding author on reasonable request.
Received: 14 July 2021; Accepted: 15 February 2022;
REFERENCES
1. Ozbudak, E. M., Thattai, M., Lim, H. N., Shraiman, B. I. & Oudenaarden, A. V.
Multistability in the lactose utilization network of Escherichia coli. Nature 427,
737–740 (2004).
2. Li, Q. et al. Dynamics inside the cancer cell attractor reveal cell heterogeneity,
limits of stability, and escape. Proc. Natl. Acad. Sci. USA 113, 2672–2677 (2016).
3. Fang, X. et al. Cell fate potentials and switching kinetics uncovered in a classic
bistable genetic switch. Nat. Commun. 9, 2787 (2018).
4. Santos-Moreno, J., Tasiudi, E., Stelling, J. & Schaerli, Y. Multistable and dynamic
CRISPRi-based synthetic circuits. Nat. Commun. 11, 2746 (2020).
5. Angeli, D., Ferrell, J. E. & Sontag, E. D. Detection of multistability, bifurcations, and
hysteresis in a large class of biological positive-feedback systems. Proc. Natl. Acad.
Sci. USA 101, 1822–1827 (2004).
6. Thomson, M. & Gunawardena, J. Unlimited multistability in multisite phosphor-
ylation systems. Nature 460, 274–277 (2009).
7. Sprinzak, D. et al. Cis interactions between notch and delta generate mutually
exclusive signaling states. Nature 465,86–90 (2010).
8. Harrington, H. A., Feliu, E., Wiuf, C. & Stumpf, M. P. Cellular compartments cause
multistability and allow cells to process more information. Biophys. J. 104,
1824–1831 (2013).
9. Craciun, G., Tang, Y. & Feinberg, M. Understanding bistability in complex enzyme-
driven reaction networks. Proc. Natl. Acad. Sci. USA 103, 8697–8702 (2006).
10. Liu, Q. et al. Pattern formation at multiple spatial scales drives the resilience of
mussel bed ecosystems. Nat. Commun. 5, 5234 (2014).
11. Bastiaansen, R. et al. Multistability of model and real dryland ecosystems through
spatial self-organization. Proc. Natl. Acad. Sci. USA 115, 11256–11261 (2018).
12. Kelso, S. Multistability and metastability: understanding dynamic coordination in
the brain. Philos. Trans. R. Soc. Lond. B Biol. Sci. 367, 906–918 (2012).
13. Gelens, L. et al. Exploring multistability in semiconductor ring lasers: theory and
experiment. Phys. Rev. Lett. 102, 193904 (2009).
14. Larger, L., Penkovsky, B. & Maistrenko, Y. Laser chimeras as a paradigm for
multistable patterns in complex systems. Nat. Commun. 6, 7752 (2015).
15. Landa, H., Schiró, M. & Misguich, G. Multistability of driven-dissipative quantum
spins. Phys. Rev. Lett. 124, 043601 (2020).
16. Pisarchik, A. N. & Feudel, U. Control of multistability. Phys. Rep. 540, 167–218
(2014).
17. Kothamachu, V. B., Feliu, E., Cardelli, L. & Soyer, O. S. Unlimited multistability and
Boolean logic in microbial signalling. J. R. Soc. Interface 12, 20150234 (2015).
18. Banaji, M. & Pantea, C. The Inheritance of Nondegenerate Multistationarity in
Chemical Reaction Networks. SIAM J. Appl. Math. 78, 1105–1130 (2018).
19. Feliu, E., Rendall, A. D. & Wiuf, C. A proof of unlimited multistability for phos-
phorylation cycles. Nonlinearity 33, 5629–5658 (2020).
20. Brown, G. & Ceredig, R. Modeling the Hematopoietic Landscape. Front. Cell Dev.
Biol. 7, 104 (2019).
21. Wheat, J. C. et al. Single-molecule imaging of transcription dynamics in somatic
stem cells. Nature 583, 431–436 (2020).
22. Orkin, S. H. & Zon, L. I. Hematopoiesis: An evolving paradigm for stem cell biol-
ogy. Cell 132, 631–644 (2008).
23. Birbrair, A. & Frenette, P. S. Niche heterogeneity in the bone marrow. Ann. N. Y.
Acad. Sci. 1370,82–96 (2016).
24. Ling, K. W. et al. Gata-2 plays two functionally distinct roles during the ontogeny
of hematopoietic stem cells. J. Exp. Med. 200, 871–872 (2004).
25. Liew, C. W. et al. Molecular analysis of the interaction between the hematopoietic
master transcription factors gata-1 and pu.1. J. Biol. Chem. 281, 28296–28306
(2006).
26. Akashi, K., Traver, D., Miyamoto, T. & Weissman, I. L. A clonogenic common
myeloid progenitor that gives rise to all myeloid lineages. Nature 404, 193–197
(2000).
S. Wu et al.
8
npj Systems Biology and Applications (2022) 10 Published in partnership with the Systems Biology Institute

27. Friedman, A. D. Transcriptional control of granulocyte and monocyte develop-
ment. Oncogene 26, 6816–6828 (2007).
28. Bresnick, E. H., Lee, H.-Y., Fujiwara, T., Johnson, K. D. & Keles, S. Gata switches as
developmental drivers. J. Biol. Chem. 285, 31087–31093 (2010).
29. Kaneko, H., Shimizu, R. & Yamamoto, M. GATA factor switching during erythroid
differentiation. Current Opinion in Hematology 17, 163–168 (2010).
30. Snow, J. W. et al. Context-dependent function of ‘GATA switchl’sites in vivo.
Blood 117, 4769–4772 (2011).
31. Olariu, V. & Peterson, C. Kinetic models of hematopoietic differentiation. Wiley
Interdiscip. Rev. Syst. Biol. Med. 11, e1424 (2019).
32. Ali Al-Radhawi, M., Del Vecchio, D. & Sontag, E. D. Multi-modality in gene reg-
ulatory networks with slow promoter kinetics. PLoS Comput. Biol. 15,1–27 (2019).
33. Jia, W., Deshmukh, A., Mani, S. A., Jolly, M. K. & Levine, H. A possible role for
epigenetic feedback regulation in the dynamics of the epithelial-mesenchymal
transition (EMT). Phys. Biol. 16, 066004 (2019).
34. Del Sol, A. & Jung, S. The importance of computational modeling in stem cell
research. Trends Biotechnol. 39, 126–136 (2020).
35. Teschendorff, A. E. & Feinberg, A. P. Statistical mechanics meets single-cell biol-
ogy. Nat. Rev. Genet. 22, 459–476 (2021).
36. Sonawane, A. R., DeMeo, D. L., Quackenbush, J. & Glass, K. Constructing gene
regulatory networks using epigenetic data. NPJ Syst. Biol. Appl. 7, 45 (2021).
37. Conde, P. M., Pfau, T., Pacheco, M. P. & Sauter, T. A dynamic multi-tissue model to
study human metabolism. NPJ Syst. Biol. Appl. 7, 5 (2021).
38. Huang, S., Guo, Y.-P., May, G. & Enver, T. Bifurcation dynamics in lineage-
commitment in bipotent progenitor cells. Dev. Biol. 305, 695–713 (2007).
39. Bokes, P., King, J. R. & Loose, M. A bistable genetic switch which does not require
high co-operativity at the promoter: a two-timescale model for the PU.1-GATA-1
interaction. Math. Med. Biol. 26, 117–132 (2009).
40. Chickarmane, V., Enver, T. & Peterson, C. Computational modeling of the hema-
topoietic erythroid-myeloid switch reveals insights into cooperativity, priming,
and irreversibility. PLoS Comput. Biol. 5, e1000268 (2009).
41. Bokes, P. & King, J. R. Limit-cycle oscillatory coexpression of cross-inhibitory
transcription factors: a model mechanism for lineage promiscuity. Math. Med .
Biol. 36, 113–137 (2019).
42. Tian, T. & Smith-Miles, K. Mathematical modeling of GATA-switching for reg-
ulating the differentiation of hematopoietic stem cell. BMC Syst. Biol. 8, S8 (2014).
43. Mojtahedi, M. et al. Cell fate decision as high-dimensional critical state transition.
PLoS Biol. 14,1–28 (2016).
44. Semrau, S. et al. Dynamics of lineage commitment revealed by single-cell tran-
scriptomics of differentiating embryonic stem cells. Nat. Commun. 8, 1096 (2017).
45. Mackey, M. C. Periodic hematological disorders: Quintessential examples of
dynamical diseases. Chaos 30, 063123 (2020).
46. Tian, T. & Burrage, K. Stochastic models for regulatory networks of the genetic
toggle switch. Proc. Natl. Acad. Sci. USA 103, 8372–8377 (2006).
47. Feng, J., Kessler, D. A., Ben-Jacob, E. & Levine, H. Growth feedback as a basis for
persister bistability. Proc. Natl. Acad. Sci. USA 111, 544–549 (2014).
48. Lebar, T. et al. A bistable genetic switch based on designable DNA-binding
domains. Nat. Commun. 5, 5007 (2014).
49. Semenov, S. N. et al. Autocataly tic, bistable, oscillatory networks of biologically
relevant organic reactions. Nature 537, 656–660 (2016).
50. Perez-Carrasco, R. et al. Combining a toggle switch and a repressilator within the
AC-DC circuit generates distinct dynamical behaviors. Cell Syst. 6,521–530.e3 (2018).
51. Maity, I. et al. A chemically fueled non-enzymatic bistable network. Nat. Commun.
10, 4636 (2019).
52. Kobayashi, H. et al. Programmable cells: Interfacing natural and engineered gene
networks. Proc. Natl. Acad. Sci. USA 101, 8414–8419 (2004).
53. Kitano, H. Biological robustness. Nat. Rev. Genet. 5, 826–837 (2004).
54. Kitano, H. Towards a theory of biological robustness. Mol. Syst. Biol. 3, 137 (2007).
55. Beaumont, M. A., Zhang, W. & Balding, D. J. Approximate bayesian computation
in population genetics. Genetics 162, 2025–2035 (2002).
56. Turner, B. M. & Van Zandt, T. A tutorial on approximate bayesian computation. J.
Math. Psychol. 56,69–85 (2012).
57. Grass, J. A. et al. GATA-1-dependent transcriptional repression of GATA-2 via
disruption of positive autoregulation and domain-wide chromatin remodeling.
Proc. Natl. Acad. Sci. USA 100, 8811–8816 (2003).
58. Tian, T. & Burrage, K. Implicit taylor methods for stiff stochastic differential
equations. Appl. Numer. Math. 38, 167–185 (2001).
59. Bechtold, B. Violin plots for Matlab. https://github.com/bastibe/Violinplot-Matlab
(2015).
60. Duddu, A. S., Sahoo, S., Hati, S., Jhunjhunwala, S. & Jolly, M. K. Multi-stability in
cellular differentiation enabled by a network of three mutually repressing master
regulators. J. R. Soc. Interface 17, 20200631 (2020).
61. Yang, L., Sun, W. & Turcotte, M. Coexistence of Hopf-born rotation and hetero-
clinic cycling in a time-delayed three-gene auto-regulated and mutually-
repressed core genetic regulation network. J. Theor. Biol. 527, 110813 (2021).
62. Velten, L. et al. Human haematopoietic stem cell lineage commitment is a con-
tinuous process. Nat. Cell Biol. 19, 271–281 (2017).
63. Hamey, F. K. & Göttgens, B. Demystifying blood stem cell fates. Nat. Cell Biol. 19,
261–263 (2017).
64. Laurenti, E. & Göttgens, B. From haematopoietic stem cells to complex differ-
entiation landscapes. Nature 553, 418–426 (2018).
65. Ackers, G. K., Johnson, A. D. & Shea, M. A. Quantitative model for gene regulation
by lambda phage repressor. Proc. Natl. Acad. Sci. USA 79, 1129–1133 (1982).
ACKNOWLEDGEMENTS
This work was supported by National Nature Scientific Foundation of P.R. China (Nos.
11931019 and 11775314).
AUTHOR CONTRIBUTIONS
T.T. conceived the main strategy. S.W. developed the method, performed the
computation and analysis. S.W, T.Z. and T.T. interpreted the results. S.W. and T.T.
drafted the manuscript. S.W. and T.T. finalised the final paper with feedback from all
authors. All authors read and approved the final version of the manuscript.
COMPETING INTERESTS
The authors declare no competing interests.
ADDITIONAL INFORMATION
Supplementary information The online version contains supplementary material
available at https://doi.org/10.1038/s41540-022-00220-1.
Correspondence and requests for materials should be addressed to Tianhai Tian.
Reprints and permission information is available at http://www.nature.com/
reprints
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims
in published maps and institutional affiliations.
Open Access This article is licensed under a Creative Commons
Attribution 4.0 International License, which permits use, sharing,
adaptation, distribution and reproduction in any medium or format, as long as you give
appropriate credit to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made. The images or other third party
material in this article are included in the article’s Creative Commons license, unless
indicated otherwise in a credit line to the material. If material is not included in the
article’s Creative Commons license and your intended use is not permitted by statutory
regulation or exceeds the permitted use, you will need to obtain permission directly
from the copyright holder. To view a copy of this license, visit http://creativecommons.
org/licenses/by/4.0/.
© The Author(s) 2022
S. Wu et al.
9
Published in partnership with the Systems Biology Institute npj Systems Biology and Applications (2022) 10