Supplementary Information
Index
S1. Bistability of Stochastic Switches
Page 1
S2. Biological Ergodicity and Derivation of Eq. (1).
Page 2
S3. Nonlinear Master Equation Including Growth, and Modeling of a Stochastic
Toggle Switch with Growth Difference
Page 4
S4. Duality in the interpretations of Ph (t )
Page 7
S5. Estimation of growth rates
Page 8
S6. Reproducibility of Fig. 1 B
Page 9
S7. Generality of nonlinear dependence of Ph (t ) on its initial condition and
growth rate difference
Page 9
S8. Estimation of switching frequencies from experimental data
Page 9
S9. Two mechanisms to realize population-level apparent bistability
Page 11
S10. Analytic estimation of the difference between the population-level and the
single-cell-level relative stability.
Page 12
Supplementary Reference
Page 14
Supplementary Figure legends
Page 15
S1. Bistability of Stochastic Switches
Individual cells are exposed to a variety of environmental fluctuations. In
addition to these external fluctuations, chemical reactions in a cell can show significant
stochasticity because of small copy numbers of molecules. It is thought that intracellular
systems show stochastic dynamics due to these external and internal stochasticities[1-6].
Because of this stochastic nature of intracellular systems, several dynamical properties
of intracellular systems must be characterized from a stochastic viewpoint [7-13].
Bistability is one of such properties. In a deterministic description, bistability
is defined as the existence of two stable equilibrium points. In case of a stochastic
description, however, this definition cannot be adopted because the switch can flip from
1
one equilibrium state to the other due to the stochasticity. At least, we have two possible
extended definitions of the bistability: one is that both equilibrium states are stable
enough to keep their states for a long time: the other is that a population of switches
shows bimodal distribution of states at a stationary state. The latter is frequently used to
investigate the bistability of intracellular switches because the distribution of states can
be measured by some experimental devices such as flow cytometry.
In this paper, we explicitly distinguish these two definitions by using a term
“relative stability” or “relative bistability”. Relative stability indicates the relative
stabilities of the two equilibrium states, and relative bistability means that the relative
stabilities of the two states are almost the same. In contrast, we use “stability” or
“absolute stability” to indicate the stability of each equilibrium state that is defined to be
the average interval during which the state can be sustained. Furthermore, as
demonstrated in the body of this paper, relative stability, relative bistability, and
absolute stability can be defined both from a population-level and single-cell-level. In
the population-level, relative stability is defined by the percentages of the states
observed by a population-level-measurement such as the flow cytometry. This
population-level relative stability is specifically denoted as “apparent relative stability”.
“Apparent relative bistability” and “apparent absolute stability” are also defined
similarly. In contrast, single-cell-level relative stability and single-cell-level absolute
stability are defined as the relative and absolute dwelling time of the switch in a state.
S2. Biological Ergodicity and Derivation of Eq. 1.
In order to evaluate the biological ergodicity, we need to model stochastic
behaviors of cells. While the stochasticity of cells can be modeled in several ways, we
assume in this paper that the stochastic behavior of a cell can be modeled as a stochastic
process [14]. Once modeled as a stochastic process, the behavior of a single cell can be
regarded as a sample path of this stochastic process, and the change in the percentage of
cells in a certain state in a population of cells can be derived, for example, with
chemical master equations. Relying on this relation between single-cell-level and
population-level descriptions, we can infer the single-cell behavior from the
2
population-level experimental data when the distribution of states of cells converges to
a certain stationary distribution. In the case of biological switches with two states, for
example, it is expected that at equilibrium the ratio of the percentages of cells in the two
states represents the relative stability of those states at the single-cell-level. In the case
of a biological switch, this identification of the equilibrium distribution with
single-cell-level stability is called “biological ergodicity” in this paper. More generally,
we define it such that the percentage of a cell population in a particular state is identical
to the probability to find a single cell in that state. We adopt this definition of biological
ergodicity because the definition is similar to a definition of the traditional ergodicity:
the probability of finding a system from a population of the systems to be in a certain
state is identical to the probability of finding a system in that same state at any given
time. Thus, this biological ergodicity is just an analogy of the traditional ergodicity.
As a result, the biological ergodicity is never related to the traditional ergodicity in
physics and thermodynamics. In this paper, however, the biological ergodicity has some
relation to the concept of the ergodicity in the theory of stochastic processes because we
model the dynamical behavior of cells with a stochastic process in this paper.
We have, however, no general way to test whether “ the biological ergodicity”
holds. While we have no general result to theoretically guarantee ergodicity, the
ergodicity of a stochastic process is guaranteed mathematically if the stochastic process
of the state of a cell is a finite-state Markov process and for arbitrary initial conditions
converges to a stationary state [15]. Since the dynamical behavior of a cell is usually
modeled with a Markov stochastic process, the existence of a unique stationary state is
typically regarded to guarantee the biological ergodicity. Some biological properties,
however, such as intercellular interactions and the different growth rates of cells in
different states prevent us from modeling a stochastic behavior of cells with a traditional
Markov process. Thus, if intercellular interactions or a difference in growth rates are not
negligible, the existence of a unique stationary state does not necessarily guarantee the
biological ergodicity. As a result, when we experimentally observe a unique stationary
state of an intracellular system, one way to validate biological ergodicity is to test if the
system can be represented as a traditional Markov process.
3
We used this procedure to show that the biological ergodicity does not
necessarily hold for one of the toggle switches. Specifically, to derive eq. 1, we assume
that the switch in each cell flips independently of the other cells and that the flipping of
the switch has the Markov property with flipping frequencies kl  h and kh  l . We also
assume that the growth rates of the cells in different states are identical. The stochastic
flipping of the switches in a population of cells can then be modeled by the traditional
Markov process with two states as in eq. 1 [15,16]. As shown in eq. 2, the stationary
solution of eq. 1 is determined by the ratio of the flipping frequencies, and thus the
relative stability of the switch can be inferred from the stationary state. In addition,
Ph (t ) and Pl (t ) in eq. 1 can be regarded not only to represent the percentages of cells
in the high and low states in a population of cells, but also to represent the probabilities
that a single cell is in the high and low states. Thus, in this sense, eq. 1 simultaneously
describes the population-level and single-cell-level behaviors of the switch when the
biological ergodicity holds.
S3. Nonlinear Master Equation Including Growth, and Modeling of a Stochastic
Toggle Switch with Growth Difference
In order to obtain eq. 3, we have to include the effect of growth rates in eq. 1.
Since eq. 1 represents the Markov process of the switch, it can be regarded formally as a
version of chemical master equation(CME). Thus, we first extend a CME to incorporate
the effect of cell growth. When the chemical master equation is used, the stochastic
dynamics of an intracellular network is typically formulated as follows:
Consider a spatially homogeneous mixture of N ( 1) kinds of molecular
species in a volume V(t), which chemically interact through M ( 1) reaction channels.
The state of the system at t is specified by a vector of random processes,
X (t )  { X1 (t ),
, X N (t )} , where X i (t ) is the number of copies of the ith molecule in
the system at time t
(i = 1, … , N), and X i (t )  Z  for i {1,
stoichiometric coefficient matrix S  ( s1 ,
, N } . Once the
sM ) , the propensity function a j ( X ) for all
reaction channels, and the initial state of the system at time t 0 , X (t0 )( x0 ) , are
specified, the stochastic time evolution of the system designated by the conditional
4
probability density function P( x, t x0 , t0 )  Prob{X (t )  x, given that X (t0 )  x0} , can
be described as a jump-type Markov process on the nonnegative N-dimensional integer
lattice with the following chemical master equation:
M

P( x, t x0 , t0 )   a j ( x  s j ) P( x  s j , t x0 , t0 )  a j ( x) P( x, t x0 , t0 )  .
t
j 1
As defined, P( x, t x0 , t0 ) represents the probability of a cell in state x(t ) at time t
with the initial state x0 (t ) at time t 0 . At the same time, however, it can be interpreted
as the percentage of cells that are in state x(t ) at time t in an infinitely large ensemble.
In other words, it describes the dynamics of a sufficiently large ensemble of genetically
identical cells. In the CME, however, the changes in the percentage of the cells in state
x(t ) are induced only by the probabilistic occurrences of chemical reactions in each
cell. Thus, it is implicitly assumed in this description that the rate at which cells grow is
the same regardless of their state x(t ) .
The effect of the cell growth can be introduced into the chemical master
equation as a nonlinear term. First we assume that an ensemble of the cells is so large
that the percentage of cells in state x(t ) can be described as a nonnegative real number
P( x, t )( P( x, t x0 , t0 )) . Let the number of cells of the ensemble at time t be E (t ) where
the dependence on t represents the changes in the size of the ensemble as the result of
cell growth. Next we derive the changes in P( x, t ) induced by the cell growth after
infinitesimally small time  t . The interaction between this change and that induced by
intracellular chemical reactions can be handled separately because of the infinitely small
t .
Thus we abbreviate the contributions of the stochastic chemical reactions in
the derivation of the effect of the cell growth. Let’s assume that a cell in state x grows
at the rate g ( x ) , where g ( x ) is a real-valued function of x . Note that a negative value
of g ( x ) means that cells in x cannot grow and die spontaneously. Then E (t   t )
and P ( x, t   t ) are given as follows:
5
E (t   t )   1  g ( x ') t  P( x ', t ) E (t ),
x'
P ( x, t   t ) 
1  g ( x) t  P( x, t ) E (t ) .
E (t   t )
By the definition of the ensemble average * , we have E (t   t )  1  g ( x ')  t  E (t ) ,
and thus
d E (t )
 g ( x(t ) E (t )
dt
holds. While the evolution of the size of the ensemble depends on the time evolution of
P ( x, t ) ,
P ( x, t   t ) does not depend on E (t ) . Instead, we have
P( x, t   t ) 
1  g ( x) t  P( x, t )
.
1  g ( x ') t  P( x ', t )
x'
Expanding this equation for  t , we get
P( x, t   t ) 
1  g ( x) t  P( x, t )
1  g ( x)  t
 1   g ( x)  g ( x)   t  P( x, t )  O( t 2 ).
Thus the time evolution of P( x, t ) is obtained as follows:
P( x, t )
P ( x, t   t )  P ( x , t )
 lim
 t 0
t
t
  g ( x)  g ( x)  P( x, t ).
As easily confirmed, if
 P( x , t )  1 , then  P( x, t | x , t )  1
0
0
0
0
holds for t  0 , and
x
x0
0  P( x, t | x0 , t0 )  1 if 0  P( x0 , t0 )  1 . Thus the above equation describes the time
evolution of probability density functions. Because the influences of stochastic chemical
reactions and the cell growth are independent here, we obtain the following nonlinear
chemical master equation (NCME) in which the cell growth is incorporated:
M

P( x, t x0 , t0 )   a j ( x  s j ) P( x  s j , t x0 , t0 )  a j ( x) P( x, t x0 , t0 )    g ( x)  g ( x)  P( x, t ) .
t
j 1
(A.1)
6
The nonlinearity of this equation becomes obvious if we replace
definition,
g ( x) with its
 g ( x) P( x, t | x , t ) . In addition, the equation is reduced to the chemical
0
0
x
master equation when g ( x ) is independent of x . The derivation of eq (3) from eq. (1)
is straightforward because the difference between them is simply the term representing
the effect of different growth rates.
A formula similar to our nonlinear CME is used to model spatially
heterogeneous chemical reactions [17]. This nonlinear master equation can also be
viewed as an extension not only of the usual master equation but also of several other
models. One such model is the cell population balance model in population dynamics.
In population dynamics, the variable is not the probability density function P( x, t ) but
the density function of cells, D ( x, t ) . Thus P( x, t ) is obtained from the population
balance model by normalizing D ( x, t ) with the total cell density. While the population
balance model is essentially equivalent to the NCME, the derivation of the NCME is
more like that of the usual CME and stochastic simulations such as Gillespie’s
algorithm. Furthermore, since P( x, t ) is defined on integer lattice, the numerical
simulation of the NCME is more straightforward than that of the population balance
model.
Another model closely related to the NCME is the replicator equation used in
population and evolutionary dynamics. Equation (A.1) can be reduced to a replicator
equation by removing the terms representing chemical reactions. In the context of the
replicator equation, these terms can be interpreted, for example, as the influence of
mutations or migration of species. We also should note that a nonlinear Fokker-Plank
model incorporating the effect of growth rate was recently proposed [18].
S4. Duality in the interpretations of Ph (t ) .
It should be noted that Ph (t ) and Pl (t )
in eq. 3 describe the percentages of
cells in the high and low state at time t in a population of cells. As mentioned, the same
interpretation of P( x, t ) is possible for eq. 1 when the biological ergodicity holds. In
addition, Ph (t ) and Pl (t ) in eq. 1 can also be interpreted as the probabilities that a
certain cell is in high and low states at time t even though the biological ergodicity does
7
not hold. Because of this dual meaning of eq. 1 under the condition that biological
ergodicity holds, we can infer the single-cell-behavior of a switch directly from the
stationary state of a population of cells when biological ergodicity holds. However, the
stationary solution of eq. 3 is not directly related to the ratio of the flipping frequencies.
As a result, the relative stability of the switch can no longer be inferred from the
stationary state of a population of cells when the growth difference is not negligible. If
we know the flipping frequencies of the switch even though the biological ergodicity
does not hold, however, the single-cell-level behaviors of switches can be represented
not by eq. 3 but by eq. 1. Thus, in this paper, we represent the single-cell-level
behaviors of the switches with eq. 1 when we succeeded in estimating the flipping
frequencies with eq. 3.
S5. Accuracy of growth rate estimation and possible origins of the growth rate
difference
In our method, cell numbers are counted directly by a flow cytemeter. In the method
with optical density (O.D.), in contrast, cell numbers are not counted directly because
O.D. measures medium turbidity. Thus, our method is expected to have at least
comparable accurate to the method with O.D.
By conducting O.D. measurements, we confirmed that the growth difference
can be detected by O.D. (data not shown). However, measurement by O.D. showed
large variation between independent measurements of the same sample. In contrast, the
variation was quite small in our method. Furthermore, our method is more suited for the
estimation of growth difference because our method measures the growth difference
directly while the method with O.D. requires us to calculate the difference from
absolute growth rates measured. It should be noted that the logic behind our method is
similar to that of avoiding computation errors in computer programming. In computer
programming, one is discouraged from obtaining a small value by subtracting a large
number from another large number and is instead encouraged to define the small value
as a variable.
The growth difference detected in our experiment have several possible causes
such as the weak toxicity of green fluorescent proteins (GFP) and biased expression
8
level of the two regulatory proteins in the toggle switch. Because the high state has
lower growth rate than the other for all plasmid tested, we think that the weak toxicity
of GFP is the most plausible origin of the growth difference.
S6. Reproducibility of Fig. 1 B
To test the reproducibility of Fig.1 B, we conducted the same experiment independently
(shown with red curve in Figure S2), and observed that the reproducibility is high
enough. We also validated the variability of measurement by a flow cytometer by
repeating the measurement three times for each condition. We found that our
measurement was quite reproducible because the coefficient of variation was 5 % on
average.
S7. Generality of nonlinear dependence of Ph (t ) on its initial condition and
growth rate difference
To verify the generality of the nonlinearity of Ph (t ) , we conducted the same
experiment as done in Fig. 1 B for another two toggle switches, pTAK132, and
pIKE107. Similarly to Fig.1 B, we observed nonlinear dependence of Ph (t ) on its
initial condition as shown in Fig. S2. The estimation of growth difference for these
toggle switches also revealed that the growth differences are 0.094 h -1 and 0.449 h -1 ,
respectively. These growth differences can also explain the nonlinear dependences of
Ph (t ) as shown in Fig. S2. This result clearly demonstrates that the break of
biological ergodicity is not a special case for pTAK131, but a general property shared
by other toggle switches.
S8. Estimation of switching frequencies from experimental data
With the estimated growth rate difference, we numerically calculate the time
evolution of eq. 3. While the growth rate difference is estimated, the switching
frequencies k l  h and k h l , are unknown. We therefore conducted simulations for
various combinations of k l  h and k h l . The results are shown in Fig. S1, where we can
easily see that the simulation results and the experimental data almost coincide when
9
both k l  h and k h l are less than 102  h 1 . This means that less than one in every 100
toggle switches flips in one hour.
Since the relation between Ph (t ) and Ph (0) is insensitive to the change in
k l  h and k h l if they are less than 102  h 1 , we need to use other information to
estimate k l  h and k h l more accurately.
First we used experimental values of Ph (t ) at the stationary state to estimate
the k l  h shown in Fig. S4. Fig. S4 is the plot of Ph (t ) at the stationary state as a
function of k l  h . From this numerical estimation in Fig. S4 and the Ph (t ) in Fig. S3,
we eventually estimate that kl  h  103  h 1 . While we implicitly assumed here that eq.
3 has a unique stationary state, it is not trivial because eq. 3 is no longer linear.
However, the existence of such a unique stationary state can be easily proved for eq. 3.
Since Pl  Ph  1, eq. 3 can be simplified as
dPh
 f ( Ph )  k hl Ph  k l h (1  Ph )  ( g h  g l )(1  Ph ) Ph .
dt
For k l h  0 and k h l  0 , we easily find that f (0) f (1)  0 . Since f ( Ph )
is quadratic with respect to Ph , we have the unique Ph* satisfying f ( Ph* )  0 . In
addition, the stationary state is stable because f (0)  0 and f (1)  0 . When k l h  0 ,
f (0)  0 holds, and thus Ph*  0 is one stationary state. Another stationary state can
exist only when g h  g l  k hl . Similarly, Ph*  1 becomes one stationary state if
k h l  0 , and the another stationary state can exist when g h  g l  k l h . For our case,
g h  g l  0 is verified experimentally. Thus, k l h  0 .In addition, because the
experimental stationary state shown in Fig. S3 is obviously different from Ph*  1 , our
analysis here is valid even if k h l  0 .
Next we estimate the upper bound of the time during which the high state can
be sustained. In our experiment we confirmed that the cells that were induced to the
high state can sustain the high state for more than 50 hours (Fig. S5). So we numerically
10
estimated Ph (t ) at t = 50 hours as a function of k h l under the assumption that all cells
are in the high state (Fig. S6). As pointed out in the text, the contamination by low-state
cells never lengthens this period. This numerical estimation thus provides us with the
upper bound of k h l . From the comparison of the experimental evidence and this
numerical simulation, we estimate that kh l  1.5 108  h 1 because the percentage of
the low-state cells is less than 0.01% when kh l  1.5  108  h 1 . We used 0.01% as a
threshold because we measured 10,000 cells to obtain each histogram.
S9. Two mechanisms to realize population-level apparent bistability
Our result indicates that the apparent bistability at the population level can be
produced not only by the bistability of the switch in the individual cells but also by the
balance between the growth rate difference and the biased stability of the switch.
To demonstrate the different parameter dependencies of these two
mechanisms, we conducted numerical simulations of eqs. 1 and 3 for different sets of
switching frequencies. As shown in Fig. S7, the two mechanisms are not consistent in
the sense that it is difficult to achieve apparent bistability with and without growth
difference with only one parameter set of flipping frequencies. Thus we have two
parameter
regions:
a
switch-dominant
bistability
region
and
a
growth-difference-dominant bistability region. This numerical observation can be
proven more rigorously and analytically. Without loosing generality, we can assume
that g  g l  g h  0 . By solving eqs. 1 and 3 at the stable stationary state, we
respectively obtain the percentages of high state cells in the population-level apparent
distribution and the single-cell-level distribution without growth as follows:
Ph 
k h l  k l h  g  (k h l  k l h  g ) 2  4k l h g
2g
~
Ph  kl h /( kl h  k hl ) .
,
(A.2)
(A.3)
~
By solving eqs. (A.2) and (A.3) for Ph  1 / 2 or Ph  1 / 2 , we can have the
conditions that the apparent and single-cell-level distributions show symmetric
11
bimodality. The conditions for (A.2) and (A.3) are respectively g  2(k l h  k hl )
and k l h  k hl  0. This result indicates that the switch without growth difference
shows symmetric bimodality when the flipping frequencies are identical while the
switches with growth difference show the apparent symmetric bimodality when the
flipping frequencies are biased to the high state and the difference is comparative to the
growth difference. As a result, this result demonstrates that the two mechanisms to
achieve the population-level symmetric bimodality, the switch-dominant and
growth-difference-dominant bistabilites, are mutually exclusive.
S10. Analytic estimation of the difference between the population-level and the
single-cell-level relative stability.
For experimental evaluation, it is important to have a criteria with which we
can estimate how the apparent bistability can differs from the single-cell-level real
stability of the switch. The criteria intuitively obtained is that the difference between the
population-level and the single-cell-level stability is large when the growth difference is
much larger than the switching frequency of the switch. Since this intuitive criteria does
not enable us to estimate the effect of growth-difference quantitatively, a rigorous
quantitative criteria is still worth calculating.
For more general situation, the difference between the apparent distribution
~
and the single-cell-level distribution without growth effect, P  Ph  Ph , can be
represented as
2

1  1  ( g   k ) 2   k
1
P   k 
2
g

where
 k  (kl h  k hl ) /( kl h  k hl )
and

,


 g  g /( k l  h  k h l ) .
From
the
definition,  1   k  1 . As easily proven, P converges to 0 as  g  0 . This
means that the difference of the apparent distribution and the single-cell-level
distribution can be neglected. The derivative of P with respect to  k is
12
d (P)
1
.
 1
d g
1   g2  2 g  k
d (P)
0
d g
holds when
 k   k   g / 2 , and d (P)  0 for  k   k and
d g
d (P)
 0 for  k   k . Thus, P is a unimodal function with respect to  k , and
d g
P( k )   g / 4 is its unique maximum. Since  1   k  1 , P reaches its
maximum  g / 4 at  k   g / 2 when  g  2 . In addition, P converges to 0 as
 k  1 when  g  1 . Since  k  1 means that the switch flips only to the high
state, this result indicates that strongly biased flipping of the switch to the high state can
overcome the effect of the higher growth rate of the low state when  g  (k l h  k h l ) .
However, for  g  1 , P converges not to 0 but to ( g  1) /  g as  k  1
while P still converges to 0 as  k  1 . This result shows that the biased flipping
to the high state can no longer overcome the influence of growth difference if
 g  ( k l  h  k h l )
.
Furthermore,
when
g  2
,
P
reaches
its
maximum ( g  1) /  g at  k  1 , and P becomes monotonously increasing function
for  1   k  1 .
In conclusion, our theoretical analysis provides us with the following criteria:
(1) when the growth difference is less than the sum of the flipping frequencies, then the
maximum difference of the stationary states is  g / 4 , but the effect of the growth
difference can be overcome by the sufficiently biased flipping to the state with the
lower growth rate.
(2) when the growth difference is greater than the sum of the flipping frequencies but is
less than the twice of the sum of the flipping frequencies, then the maximum
13
difference of the stationary states is  g / 4 , but the effect of the growth difference
can no longer be overcome but can be reduced by the biased flipping to the state
with lower growth rate.
(3) when the growth difference is greater than the twice of the sum of the flipping
frequencies, the growth difference completely dominates the flipping of the switch,
and thus the deviation between the apparent distribution with growth and the real
distribution without growth increases as the flipping is biased to the state with the
lower growth rate.
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