Supplementary Notes
The discrete and stochastic nature of chemical and biological processes has now been
widely recognized and substantiated.1-8 Furthermore, it has also been shown that the nondeterministic chemical master equation (CME), which describes the evolution of the
system state probability distribution, offers the more accurate description of the
underlying molecular dynamics than that provided by the classical deterministic
(bio)chemical kinetic (CCK) equations of motion. That is, CCK deterministically
describes the dynamics of such (continuous) quantities as molecular concentrations,
while CME pertains to the temporal evolution of the (discrete) system state probability
distribution that arises because of the random nature of microscopic molecular reaction
events. In this work we rigorously examine and comprehensively characterize the causes
of deviant non-classical effects in molecular reaction pathways and demonstrate that –
contrary to some widely held assumptions – in the presence of bimolecular and
irreversible reactions deviant effects can arise in otherwise rather ordinary systems
regardless of their size and/or complexity. We also show that under the circumstances
often present in biological molecular networks that have multiple stable operating ranges,
the underlying system behavior may dramatically diverge from that of the most likely
system trajectory and its classical kinetic description. Finally, we use the average
trajectory – which may provide a more accurate description of such deviant system
dynamics – to present an approach for empirically identifying these effects in various
natural or synthetic molecular pathways.
S1
While fundamental reasons for expecting the average to be a more comprehensive
description of system behavior are outside the scope of this paper, it would suffice to
point out that the average is an integral property of the probability distribution, while the
mode is differential. Therefore, the average should, generally, be expected to result in an
intrinsically more inclusive measure of the system stochastic properties. This may indeed
be seen in the context of the Type I example considered. While we have shown there that
the CME average provides an accurate description of system dynamics (and the mode or
CCK do not), it may be of further use when looking to estimate the time-scale required
for switching between the transient CCK and absolutely stable CME stationary states.
This question might be important for practical applications of this type of analysis to
biological molecular systems, as the time in question may rapidly increase with system
size. Though detailed analysis of the issue is beyond this paper, the switching time-scale in
question is captured here in the behavior of the dynamics of the average. More generally
the issue of stationary state transitions is frequently considered within the context of the
CME-based ‘mean first passage time’ framework,9 which effectively substantiates the
non-classical nature of the effects in question. Note further that the differences between
the behavior of the average and the behavior of the mode should not be confused with
the differences between the behavior of the ensemble average and the behavior of a
typical ensemble member. Differences between the latter could arise in connection with
reaction systems that, for instance, execute limit cycle oscillations, whereby the ensemble
average eventually stops oscillating while all of ensemble members continue doing so.10
Here, we use system (1) as a general mechanism for studying deviant effects since it
possesses the characteristic properties identified earlier, such as the combination of
S2
bimolecular/homodimeric, m  2 , and autoactivation/autocatalytic irreversible reaction
dynamics. To sustain the non-classical behaviors of interest these parsimonious features
either need to be directly present in the pathway, e.g. as a homodimeric reaction in (1), or
equivalently generated via certain extended stoichiometry. For example, a heterodimeric
UV
W could have an effectively homodimeric dynamics when it is
reaction U  V k
kV , kU
V and
composed with otherwise rapidly interconverting monomers, i.e. U 
kU , kV  kUV , or if any similar mechanism rapidly equilibrates these monomer
components within the system.11 As shown in this work, instantiations of just this one
reaction scheme (1) are capable of exhibiting a variety of deviant effects, including the
explicitly discussed Type I and II. A futile enzymatic cycle system used as an example of
a Type III effect can also be put within the context of (1), whereby its homodimeric
reaction is replaced by a heterodimeric one as discussed above.12
Type I Effects
These effects arise due to the failure of the leading order approximation to the underlying
CME probability distribution, as we have shown for system (2). Additional details of this
analysis are presented next.
Classical Approach: The CCK ODE equations for system (2) can be written directly as
dx
 k 2 xy  2k1 x 2 ;
dt
dy
 2k1 x 2  k 2 xy .
dt
These can be further simplified (using mass conservation x  y  xT ) to give
S3
(S1)
dx
 ( k 2 xT ) x  ( k 2  2 k 1 ) x 2 ,
dt
(S2)
which yields the expressions for the positions of the classical steady states as:
x1ss  0 & x2ss  xT k 2 (k 2  2k1 ) .
(S3)
Upon checking the relative stability of these states via linear perturbation of equation
(S2), we can ascertain that classically x1ss is unstable, while x 2ss is stable.
Non-Classical Approach: Temporarily disregarding CCK results and considering the
non-classical behavior of the system, we notice that here X is produced from Y
autocatalytically. Thus, should X somehow be depleted via the first reaction channel,
there is no way of producing it any more. Since biochemical reactions are stochastic by
nature, there is always a chance that this would happen in a finite nontrivial system if we
wait long-enough, and once it does – the system has no way of escaping this state of no
X . That is, a simple qualitative approach would seem to indicate that – contrary to the
conclusions of the classical analysis above – it is the X 1ss  0 state that is, in fact,
absolutely stable (absorbing) and not otherwise.
For this simple system, the heuristic arguments above can be further validated by the
explicit master equation analysis. As, via (6),
P( X ; t )
 c1 ( X  2)( X  1) P( X  2; t ) 2  c2 ( X  1)( X T  X  1) P( X  1; t )
t
 (c1 X ( X  1) 2  c2 X ( X T  X )) P( X ; t )
S4
(S4)
is the master equation for system (2), where ci ’s are the microscopic reaction rates and
X T  X  Y . Its stationary solution, P( X , t  ) t  0 with Ps ( X )  P( X , t  ) ,
can be found for X T  1 by iteratively solving the stationary Equation (S4) at each X
state:
State
Iteration 1
Iteration 2
Iteration 3
X  0 P(0, ) / t  0  c1 Ps (2)  Ps (2)  0
X 1
X 2
(S5)
0  3c1 Ps (3)  c2 ( X T  1) Ps (1)  Ps (3)  0
0  6c1 Ps (4)  c2 ( X T  1) Ps (1)
(c1  2c2 ( X T  2)) Ps (2)  Ps (1), Ps (4)  0
X 3
Carrying on the procedure yields a confirmation of our earlier qualitative supposition,
i.e.:
Ps ( X )  P( X ; t  )   X , 0 .
(S6)
Importantly, as the origins of such Type I effects are fundamentally different from the
more commonly considered Type II (exemplified next), the fact that the CME stationary
state is { X ssCME , YssCME }  {0, X T } here, should not confuse these two types – both because
this system property remains substantially unaffected by any molecular-scale changes in
rates or initial conditions, e.g. {x ssCCK , y ssCCK }  {xT k 2 (k 2  2k1 ), xT 2k1 (k 2  2k1 )} stable
CCK steady state is, generally, different from the CME one on a system-size scale
X ssCCK  X ssCME ~ X T , and also because no system species may have a low CCK state.
S5
Type II Effects
As was cited elsewhere in this work, some of the best-studied molecular pathways are
known to be mediated by certain low abundance species and may thus be susceptible to
the deviant effects of Type II. In particular, it has been well-understood that even very
simple molecular reaction mechanisms, such as an enzyme with production and decay
rates set to be very slow compared to its activity – so that they produce a CCK steady
state of only half a molecule – will generally show some non-classical behavior effects.
(This is due to the underlying discrete nature of molecular counts, which in this instance
would mostly be zero or one, causing the system to exhibit dynamic switching between
those two states with corresponding oscillations in activity – a behavior deviant from the
CCK predictions because of a Type II(a) effect, namely the breakdown in the continuous
concentration approximation that allows for the non-integer states.) It also highlights the
general way in which such locally deviant dynamics could propagate on via biological
pathways from the low-abundance species (e.g. enzyme) to qualitatively influence the
behavior of other high-abundance ones (e.g. substrate) as noted earlier.
While generally attributed to systems with low molecular copy number species, we have
shown that in certain cases the Type II discrete effects may not require such restrictions.
Another source of such high copy number Type II effects could arise in, for example,
“bursty” dynamic processes such as predicted and demonstrated for gene expression.1, 1315
Therein, as translation is often much faster than transcription and one mRNA typically
produces many proteins – protein number increases in effectively semi-discrete bursts,
potentially placing this mechanism outside the classical limit regardless of any and all
additional stochastic effects or the number of DNA templates available. Notably, the
S6
molecular dynamics of reaction processes, such as transcription and translation, are
sometimes discussed in terms of “delays”. In the context of this work, however, the
apparent delays in transcription and translation arise merely due to the phenomenological
approximations of multiple nonlinear elementary reaction steps that individually fall well
within the scope and are indeed the focus of our study.
While simple in origin, such non-classical behavioral patterns could give rise to a further
variety of other dynamical effects as is easy to infer using our approach. They can also be
used to highlight the more subtle differences between Type II(a) and Type II(b) effects.
For instance, the addition of reaction
E
k




k
E* ,
(S7)
where E * is an inactive (e.g. phosphorylated or dephosphorylated) form of the enzyme,
will cause the production rate of system (3) to oscillate when the initial number of enzyme
molecules is odd – pulsing from zero to a fixed level and back due to Type II(a) effect in
a manner similar to that of a linear reaction discussed earlier (results not shown) as the
last molecule is alternatively activated or deactivated.
On the other hand, the presence of such additional processes as
EE
k


E  E*
S7
(S8)
– instead of (S7) – in (3) would result in a system that could randomly end up in one of
the two characteristic scenarios, described above and in Figure 4, regardless of the initial
number of enzyme molecules. That is, production of P will either shut down by running
out of the enzyme or will continue indefinitely until the substrate runs out because of one
permanently active enzyme molecule – similarly to the behavior of system (3). However,
while in system (3) the choice between these two characteristic behaviors is dictated by
the initial state of E being either odd or even – in the combined system (3) and (S8) the
choice is random with the exact probabilities of either scenario dictated by the respective
reaction rates. Note that, in contrast to (S7), here the characteristic behavior is driven by a
Type II(b) effect that arises due to the breakdown in the rate approximation, which thus
prevents the removal of the last molecule from the system (also see Methods).
The two alternatives described above in (S7) and (S8) also serve as contrasting examples
of the possible orbit behavior of these stochastic systems – one dynamically bistable and
the other simply bimodal. Of course, as noted earlier, many other combinations of such
reactions and/or initial conditions could be similarly evinced – with the discrete single
molecule-scale effects in one species causing substantial deviation from the classical
behavior in other abundantly present ones – thus potentially leading to the induction of
distinct non-classical behaviors in the overall biomolecular reaction networks where these
reactions are imbedded.
Conclusions
In this work we have considered the nature of the classical chemical kinetics (CCK)
description for (bio)chemical reaction systems as a limit of the more accurate chemical
S8
master equation (CME) formalism. The required approximations result in the potential
qualitative (and quantitative) differences between the characteristic system behaviors
elucidated via CCK and those predicted by the CME description. These, in turn, may
lead to decidedly non-classical behaviors even in otherwise ordinary-looking systems
where such approximations break down. Here we have aimed to identify such deviant
effects by the approximations taken, and to thus characterize the fundamental failure
modes of the CCK formalism (as compared to CME). As a result, we have shown that
possible non-classical effect sources could be exhaustively partitioned into three basic
types.† This – among other things – has allowed us to further establish that, while it has
been well understood that significant behavioral deviations away from the CCK model
could arise in molecular systems with very ‘low’ classical steady states of some species,
this is not a general requirement. In fact, we were able to highlight such failure modes in
what otherwise appear to be overtly ordinary systems with ‘high’ CCK steady state levels
– the only overriding requirement being that they include nonlinear (bimolecular) and
irreversible reactions.
Given the dramatic differences shown to potentially appear between CCK model
predictions and those exhibited by the underlying CME mechanism, it is important to
understand not only the origins of such effects, but also how they could be practically
predicted and identified – whether in standalone mechanisms or, particularly, when
†
It is important to note that our Type I-III characterization of deviant effects in molecular reaction
pathways is indeed exhaustive. In brief, this could be shown as follows: Any deviant effect – i.e. a CME
system behavior deviant from CCK predictions – could be characterized with this approach. If it could not
be characterized – this would mean that none of the approximations were broken, which would imply that
CCK was able to accurately describe the CME system. But the latter would indicate that no non-classical
effects were present in the system in the first place, which asserts the point.
S9
imbedded in larger biomolecular reaction networks. Based on the examples considered
herein, one practical method pertinent to biological and biomedical applications might be
to look for any substantial divergence between the behavior of an empirical system mode
(or the numerically computed deterministic CCK ODE solution) and that of the measured
average (or that obtained through kinetic Monte Carlo CME simulations via the Gillespie
Algorithm16-19). The two should not show any appreciable differences, and if they do –
the system would be likely to exhibit deviant effects (Figures 3-5). Notably, in these
cases the average may provide the more accurate/relevant deterministic system behavior
description alternative to CCK.
Also, a question may naturally arise as to the relevance of such deviant mechanism for
real in vivo and in vitro biomolecular systems and processes. On the one hand, while the
discussed discrete non-classical effects of Type II may appear chemically irrelevant, it is
nonetheless true that many key biological processes are controlled by the low molecular
count species. Therefore, it might be of interest if an important part of any non-classical
behavior in these systems is indeed contributed by the effectively discrete nature of these
processes rather than their stochastic dynamics per se. On the other hand, as discussed
here, Type I and III non-classical effects do not, generally, have similar low molecular
restrictions and may thus appear in otherwise seemingly generic biomolecular systems.
In summary, the simplicity as well as the overt ubiquity of such mechanisms driven by
bimolecular and irreversible processes in biological systems further underscore the need
for considering the possibility of deviant non-classical effects within and their influence
on the overall structure, function and evolution of any natural or synthetic pathway. Thus,
S10
this also emphasizes the importance of being able to select the appropriate continuousdeterministic, discrete-stochastic or other technologies and methodologies in practical
biotechnological and biomedical applications, such as when using kinetic methods to
predict efficiency of a biomolecular synthesis or efficacy of a pharmaceutical treatment.
S11
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