Stability and noise in biochemical switches 
William Bialek 
NEC Research Institute 
4 Independence Way 
Princeton, New Jersey 08540 
bialek res ea rch. nj. nec. cora 
Abstract 
Many processes in biology, from the regulation of gene expression in 
bacteria to memory in the brain, involve switches constructed from 
networks of biochemical reactions. Crucial molecules are present in 
small numbers, raising questions about noise and stability. Analysis 
of noise in simple reaction schemes indicates that switches stable for 
years and switchable in milliseconds can be built from fewer than 
one hundred molecules. Prospects for direct tests of this prediction, 
as well as implications, are discussed. 
I Introduction 
The problem of building a reliable switch arises in several different biological con- 
texts. The classical example is the switching on and off of gene expression during 
development [1], or in simpler systems such as phage  [2]. It is likely that the cell 
cycle should also be viewed as a sequence of switching events among discrete states, 
rather than as a continuously running clock [3]. The stable switching of a specific 
class of kinase molecules between active and inactive states is believed to play a role 
in synaptic plasticity, and by implication in the maintenance of stored memories 
[4]. Although many details of mechanism remain to be discovered, these systems 
seem to have several common features. First, the stable states of the switches are 
dissipative, so that they reflect a balance among competing biochemical reactions. 
Second, the total number of molecules involved in the construction of the switch is 
not large. Finally, the switch, once flipped, must be stable for a time long compared 
to the switching time, perhaps for development and for memory even for a time 
comparable to the life of the organism. Intuitively we might expect that systems 
with small numbers of molecules would be subject to noise and instability [5], and 
while this is true we shall see that extremely stable biochemical switches can in fact 
be built from a few tens of molecules. This has interesting implications for how we 
think about several cellular processes, and should be testable directly. 
Many biological molecules can exist in multiple states, and biochemical switches 
use this molecular multistability so that the state of the switch can be read out' by 
sampling the states (or enzymatic activities) of individual molecules. Nonetheless, 
these biochemical switches are based on a network of reactions, with stable states 
that are collective properties of the network dynamics and not of any individual 
molecule. Most previous work on the properties of biochemical reaction networks 
has involved detailed simulation of particular kinetic schemes [6], for example in 
discussing the kinase switch that is involved in synaptic plasticity [7]. Even the 
problem of noise has been discussed heuristically in this context [8]. The goal in 
the present analysis is to separate the problem of noise and stability fi'om other 
issues, and to see if it is possible to make some general statements about the limits 
to stability in switches built from a small number of molecules. This effort should 
be seen as being in the same spirit as recent work on bacterial cherootaxis, where 
the goal was to understand how certain features of the computations involved in 
signal processing can emerge robustly from the network of biochemical reactions, 
independent of kinetic details [9]. 
2 Stochastic kinetic equations 
Imagine that we write down the kinetic equations for some set of biochemical re- 
actions which describe the putative switch. Now let us assume that most of the 
reactions are fast, so that there is a single molecular species whose concentration 
varies more slowly than all the others. Then the dynamics of the switch essentially 
are one dimensional, and this simplification allows a complete discussion using stan- 
dard analytical methods. In particular, in this limit there are general bounds on 
the stability of switches, and these bounds are independent of (incompletely known) 
details in the biochemical kinetics. It should be possible to make progress on multi- 
dimensional versions of the problem, but the point here is to show that there exists 
a limit in which stable switches can be built from small numbers of molecules. 
Let the number of molecules of the 'slow species' be n. All the different reactions 
can be broken into two classes: the synthesis of the slow species at a rate f(n) 
molecules per second, and its degradation at a rate g(n) molecules per second; the 
dependencies on n can be complicated because they include the effects of all other 
species in the system. Then, if we could neglect fluctuations, we would write the 
effective kinetic equation 
dn 
dt : f(n) - g(n). (1) 
If the system is to function as a switch, then the stationarity condition f(n): g(n) 
must have multiple solutions with appropriate local stability properties. 
The fact that molecules are discrete units means that we need to give the chemical 
kinetic Eq. (1) another interpretation. It is the mean field approximation to a 
stochastic process in which there is a probability per unit time f(n) of making the 
transition n - n+ 1, and a probability per unit time g(n) of the opposite transition 
n - n - 1. Thus if we consider the probability ?(n, t) for there being n molecules 
at time t, this distribution obeys the evolution (or 'master') equation 
- f(n- 1)P(n - 1,t) + g(n + 1)P(n + 1,t)- If(n) + g(n)]P(n, t),(2) 
with obvious corrections for n: 0, 1. We are interested in the effects of stochasticity 
for n not too small. Then 1 is small compared with typical values of n, and we can 
approximate P(n,t) as being a smooth function of n. We can expand Eq. (2) in 
derivatives of the distribution, and keep the leading terms: 
OP(n, t) O{ 10 
at : o- [g(n) - f(n)]P(n,t) + --[f(n) + g(n)]P(n,t) . (3) 
This is analogous to the diffusion equation for a particle moving in a potential, but 
this analogy works only if allow the effective temperature to vary with the position 
of the particle. 
As with diffusion or Brownian motion, there is an alternative to the diffusion equa- 
tion for P(n, t) and this is to write an equation of motion for n(t) which supplements 
Eq. (1) by the addition of a random or Langevin force (): 
dn 
-- f(n) - g(n) + (t), (4) 
dt 
((t)(t')) -- If(n) + g(n)]d(t- t'). (5) 
From the Langevin equation we can also develop the distribution functional for 
the probability of trajectories n(t). It should be emphasized that all of these ap- 
proaches are equivalent provided that we are careful to treat the spatial variations 
of the effective temperature [10].  In one dimension this complication does not im- 
pede solving the problem. For any particular kinetic scheme we can compute the 
effective potential and temperature, and kinetic schemes with multiple stable states 
correspond to potential functions with multiple minima. 
3 Noise induced switching rates 
We want to know how the noise term destabilizes the distinct stable states of the 
switch. If the noise is small, then by analogy with thermal noise we expect that there 
will be some small jitter around the stable states, but also some rate of spontaneous 
jumping between the states, analogous to thermal activation over an energy barrier 
as in a chemical reaction. This jumping rate should be the product of an ;attempt 
frequency" of order the relaxation rate in the neighborhood of one stable state 
and a ;Boltzmann factor" that expresses the exponentially small probability of 
going over the barrier. For ordinary chemical reactions this Boltzmann factor is 
just exp(-F/kBT), where F  is the activation free energy. If we want to build 
a switch that can be stable for a time much longer than the switching time itself, 
then the Boltzmann factor has to provide this large ratio of time scales. 
There are several ways to calculate the analog of the Boltzmann factor for the 
dynamics in Eq. (4). The first step is to make more explicit the analogy with 
BrownJan motion and thermal activation. Recall that BrownJan motion of an over- 
damped particle is described by the Langevin equation 
dx 
-- -v'/x) + v/l), 
where ? is drag coefficient of the particle, V(x) is the potential, and the noise force 
has correlations {r/(t)r/(t)) = 2?Td(t- t), where T is the absolute temperature 
measured in energy units so that Boltzmann's constant is equal to one. Comparing 
with Eq. (4), we see that our problem is equivalent to a particle with ? = 1 
in an effective potential Vff(n) such that VJff(n) : g(n)- f(n), at an effective 
temperature Tef(n): If(n) + g(n)]/2. 
If the temperature were uniform then the equilibrium distribution of n would be 
Peq(n) oc exp[-Vff(n)/Tff]. With nonuniform temperature the result is (up to 
In a review written for a biological audience, MeAdams and Arkin [11] state that 
Langevin methods are unsound and can yield invalid predictions precisely for the case 
of bistable reaction systems which interests us here; this is part of their argument for 
the necessity of stochastic simulation methods as opposed to analytic approaches. Their 
reference for the failure of Langevin methods [12], however, seems to consider only Langevin 
terms with constant spectral density, thus ignoring (in the present language) the spatial 
variations of effective temperature. For the present problem this would mean replacing 
the noise correlation function If(n) + g(n)]6(t- t') in Eq. (5) by Q6(t- t') where (2 is 
a constant. This indeed is wrong, and is not equivalent to the master equation. On the 
other hand, if the arguments of Refs. [11, 12] were generally correct, they would imply that 
Langevin methods could not used for the description of BrownJan motion with a spatially 
varying temperature, and this would be quite a surprise. 
weakly varying prefactors) 
One way to identify the Boltzmann factor for spontaneous switching is then to 
compute the relative equilibrium occupancy of the stable states (n0 and n) and the 
unstable "transition state" at n,. The result is that the effective activation energy 
for transitions from a stable state at n = n0 to the stable state at n = n > n0 is 
F*(no - n) : 2kBT dng(n ) - f(n) 
o g(n)+f(n)' 
(9) 
where n. is the unstable point, and similarly for the reverse transition, 
. g(n)+f(n)' 
(lO) 
An alternative approach is to note that the distribution of trajectories n() includes 
locally optimal paths that carry the system from each stable point up to the tran- 
sition state; the effective activation free energy can then be written as an integral 
along these optimal paths. The use of optimal path ideas in chemical kinetics has 
a long history, going back at least to Onsager. A discussion in the spirit of the 
present one is Ref. [13]. For equations of the general form 
dn 
d : -Vdff(n) + (), (11) 
with (()(')}: 2Tff ()8(-'), the probability distribution for trajectories P[n()] 
can be written as [10] 
If the temperature Tff is small, then the trajectories that minimize the action 
should be determined primarily by minimizing the first term in Eq. (13), which is 
 1/Tf. Identifying the effective potential and temperature as above, the relevant 
term is 
1 / [- f(n) +g(n)] 2 
 di f(n) + g(n) 
We are searching for trajectories which take n() from a stable point n0 where 
f(n0) = g(n0) through the unstable point n, where f and g are again equal but the 
derivative of their difference (the curvature of the potential) has changed sign. For 
a discussion of the analogous quantum mechanical problem of tunneling in a double 
well, see Ref. [14]. First we note that along any trajectory from n0 to n, we can 
simplify the third term in Eq. (14): 
f(n) - g(n) * dnf(n ) - g(n) 
(15) 
This term thus depends on the endpoints of the trajectory and not on the path, and 
therefore cannot contribute to the structure of the optimal path. In the analogy 
to mechanics, the first two terms are equivalent to the (Euclidean) action for a 
particle with position dependent mass in a potential; this means that along extremal 
trajectories there is a conserved energy 
i 'h 2 i If(n) - 
E: - - (16) 
2 f(n) + g(n) 2 f(n) + g(n) 
At the endpoints of the trajectory we have h: 0 and f(n): g(n), and so we are 
looking for zero energy trajectories, along which 
h(t): +[f(n)) - g(nU))]. 
(17) 
Substituting back into Eq. (14), and being careful about the signs, we find once 
again Eq's. (9,10). 
Both the 'transition state' and the optimal path method involve approximations, 
but if the noise is not too large the approximations are good and the results of 
the two methods agree. Yet another approach is to solve the master equation (2) 
directly, and again one gets the same answer for the switching rate when the noise 
is small, as expected since all the different approaches are all equivalent if we make 
consistent approximations. It is much more work to find the prefactors of the rates, 
but we are concerned here with orders of magnitude, and hence the prefactors aren't 
so important. 
4 Interpretation 
The crucial thing to notice in this calculation is that the integrands in Eq's. (9,10) 
are bounded by one, so the activation energy (in units of the thermal energy/cuT) 
is bounded by twice the change in the number of molecules. Translating back to 
the spontaneous switching rates, the result is that the noise driven switching time 
is longer than the relaxation time after switching by a factor that is bounded, 
spontaneous switching time 
relaxation time 
< exp(An), (18) 
where An is the change in the number of molecules required to go fi'om one stable 
'switched' state to the other. Imagine that we have a reaction scheme in which the 
difference between the two stable states corresponds to roughly 25 molecules. Then 
it is possible to have a Boltzmann factor of up to exp(25)  10 . Usually we think 
of this as a limit to stability: with 25 molecules we can have a Boltzmann factor of no 
more than  10 . But here I want to emphasize the positive statement that there 
exist kinetic schemes in which just 25 molecules would be sufficient to have this level 
of stability. This corresponds to years per millisecond: with twenty five molecules, 
a biochemical switch that can flip in milliseconds can be stable for years. Real 
chemical reaction schemes will not saturate this bound, but certainly such stability 
is possible with roughly 100 molecules. The genetic switch in  phage operates with 
roughly 100 copies of the repressor molecules, and even in this simple system there 
is extreme stability: the genetic switch is flipped spontaneously only once in 105 
generations of the host bacterium [2]. Kinetic schemes with greater cooperativity 
get closer to the bound, achieving greater stability for the same number of molecules. 
In electronics, the construction of digital elements provides insulation against fluc- 
tuations on a microscopic scale and allows a separation between the logical and 
physical design of a large system. We see that, once a cell has access to several tens 
of molecules, it is possible to construct 'digital' switch elements with dynamics that 
are no longer significantly affected by microscopic fluctuations. Furthermore, weak 
interactions of these molecules with other cellular components cannot change the 
basic 'states' of the switch, although these interactions can couple state changes to 
other events. 
The importance of this 'digitization' on the scale of 10 - 100 molecules is illustrated 
by different models for pattern formation in development. In the classical model 
due to Turing, patterns are expressed by spatial variations in the concentration of 
different molecules, and patterns arise because uniform concentrations are rendered 
unstable through the combination of nonlinearities in the kinetics with the different 
diffusion constants of different substances. In this picture, the spatial structure of 
the pattern is linked directly to physical properties of the molecules. An alternative 
that each spatial location is labelled by a set of discrete possible states, and patterns 
evolve out of the 'automaton' rules by which each location changes state in relation 
to the neighboring states. In this picture states and rules are more abstract, and the 
dynamics of pattern formation is really at a different level of description from the 
molecular dynamics of chemical reactions and diffusion. Reliable implementations of 
automaton rules apparently are accessible as soon as the relevant chemical reactions 
involve a few dozen molecules. 
Biochemical switches have been reconstituted in vitro, but I am not aware of any 
attempts to verify that stable switching is possible with small numbers of molecules. 
It would be most interesting to study model systems in which one could confine and 
monitor sufficiently few molecules that it becomes possible to observe spontaneous 
switching, that is the breakdown of stability. Although genetic switches have cer- 
tain advantages, even the simplest systems would require full enzymatic apparatus 
for gene expression (but see Ref. [16] for recent progress on controllable in vitro 
expression systems)? Kinase switches are much simpler, since they can be con- 
structed from just a few proteins and can be triggered by calcium caged calcium 
allows for an optical pulse to serve as input. 
At reasonable protein concentrations, 10 - 100 molecules are found in a volume of 
roughly 1 (/m) 3. Thus it should be possible to fabricate an array of 'cells' with 
linear dimensions ranging from 100 nm to 10 /m, such that solutions of kinase 
and accessory proteins would switch stably in the larger cells but exhibit instability 
and spontaneous switching in the smaller cells. The state of the switch could be 
read out by including marker proteins that would serve as substrates of the kinase 
but have, for example, fluorescence lines that are shifted by phosphorylation, or by 
having fluorescent probes on the kinase itself; transitions of single enzyme molecules 
should be observable [15]. 
A related idea would be to construct vesicles containing ligand gate ion channels 
which can conduct calcium, and then have inside the vesicle enzymes for synthesis 
and degradation of the ligand which are calcium sensitive. The cGMP channels of 
rod photoreceptors are an example, and in rods the cyclase synthesizing cGMP is 
calcium sensitive, but the sign is wrong to make a switch [17]; presumably this could 
solved by appropriate mixing and matching of protein components from different 
cells. In such a vesicle the different stable states would be distinguished by different 
2Note also that reactions involving polymer synthesis (mRNA from DNA or protein 
from mRNA) are not 'elementary' reactions in the sense described by Eq. (2). Synthesis 
of a single mRNA molecule involves thousands of steps, each of which occurs (conditionally) 
at constant probability per unit time, and so the noise in the overall synthesis reaction is 
very different. If the synthesis enzymes are highly processire, so that the polymerization 
apparatus incoporates many monomers into the polymer before 'backing up' or falling off 
the template, then synthesis itself involves a delay but relatively little noise; the dominant 
source of noise becomes the assembly and disassembly of the polymerization complex. 
Thus there is some subtlety in trying to relate a simple model to the complex sequence 
of reactions involved in gene expression. On the other hand a detailed simulation is 
problematic, since there are so many different elementary steps with unknown rates. This 
combination of circumstances would make experiments on a minimal, in vitro genetic 
switch espcially interesting. 
levels of internal calcium (as with adaptation states in the rod), and these could 
be read out optically using calcium indicators; caged calcium would again provide 
an optical input to flip the switch. Ainusingly, a close packed array of such vesicles 
with -- 100 nm dimension would provide an optically addressable and writable 
memory with storage density comparable to current RAM, albeit with much slower 
switching. 
In summary, it should be possible to build stable biochemical switches from a few 
tens of molecules, and it seems likely that nature makes use of these. To test our 
understanding of stability we have to construct systems which cross the threshold for 
observable instabilities, and this seems accessible experimentally in several systems. 
Acknowledgments 
Thanks to M. Dykman, J. J. Hopfield, and A. J. Libchaber for helpful discussions. 
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