Noise and Fluctuations: An Introduction (Dover Books on Physics)
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For simplicity, we shall assume in our model the total number of molecules of each species to be conserved, this assumption defining the following conservation laws:. In view of these conservation laws, the system has only one free variable, in the following assumed to be T. The average behaviour for this model can be studied by considering the associated rate equation for the concentration 6.
We start by writing the steady state solution of the rate equation for the case of two species that bind and unbind, with constant total concentration for each species. The rate equation for the concentration of T is:. By making use of the conservation laws for concentrations that trivially follow from 2 and 3 for constant volumes, the above equation can be easily solved at the steady state giving:.
This deterministic approximation can be used to investigate the average behaviour of the number of free target molecules. By rescaling eq. As discussed in 6 , the titrative interaction induces a threshold-like behaviour on the mean number of free molecules, upon the variation of their total amount see Fig. The model considered here presents this feature with minimal ingredients, given by the binding and unbinding reactions between T and S and the conservation laws.
Indeed, when the system is in the regime in which the total amount of target molecules is smaller than the sequestrant one, almost all of them are bound in complex and their mean free amount is close to zero. We name this region the repressed regime. Conversely, in the regime where the total amount of target molecules is larger than the sequestrant one, their mean number increases linearly with T T , since the number of free molecules of the sequestrant is almost zero.
We name this region the unrepressed regime. The position of the threshold is then located close to the equimolarity point, i. As a direct outcome of this behaviour, the system becomes ultrasensitive in proximity to the threshold. This means that around this point, a small fold-change variation in the total number of target molecules can result in a large fold-change of their mean free amount.
In this specific framework, the steepness of the threshold is determined by the dissociation constant K d. It follows that a small K d implies a large affinity between the molecules, therefore leading to a sharper threshold. In the limit of infinitely large affinity between the sequestrant and the target, the system is continuous at the threshold but displays a discontinuous derivative. In this respect, the usage of the term ultrasensitivity depicts a scenario that differs from the ones typical of Goldbeter-Koshland, or cooperative Hill models 22 which, in the limit of infinite cooperation are discontinuous.
Very often, in biochemical systems, the numbers of individual molecules at play is low.
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Fluctuations around the average behaviour described by the rate equation become then relevant and, to correctly characterise these systems, it is mandatory to take into account their stochastic nature In order to do that, we seek the explicit form of the probability distribution of the number of molecules in the system at a given time. Recalling that this model has only one independent variable, let us define P T , t as the probability of observing T free molecules at time t.
The dynamics of this probability distribution is Markovian and obeys the following chemical master equation 23 , 24 :. Using the conservation laws, eqs 2 and 3 , the master equation can be written as:. Since we have a single independent variable and all reactions are reversible, the detailed balance condition is satisfied. Then, at the steady state, equilibrium is reached and there are no probability flows between the states of the system. An important feature of the system is that the possible range of the number of free T depends both on the total number of available target molecules T T and on the total number of sequestrant molecules S T.
To obtain the steady-state solution P T to the master equation, one can recursively use the detailed balance condition. Some examples of this probability distribution for different values of the dissociation constant and T T are reported in Fig. For the explored parameters range, the target distribution was always unimodal.
The various moments of the distributions can be obtained exactly by taking ensemble averages over the probability distribution P T from eq. The mean of T can be written in terms of Hypergeometric functions 25 , 26 as:. Its behaviour is plotted in Fig. In the previous section we focussed on the fluctuations originated by the discrete nature of molecules and the intrinsic randomness of their interactions. By introducing fluctuations in the total number of sequestrant molecules S T , we here investigate the influence of extrinsic noise on the present system.
Let us consider different copies of our system with different S T , randomly assigned. This mimics, for instance, the scenario in heterogeneous cell populations randomly sampled. The marginal probability distribution of free target molecules T over the different values of S T , P T , is now affected by the fluctuations in S T and differs from the one given in eq. Nonetheless, it can be expressed by making use of the law of total probability 28 , i. Given this scheme, let us now study the consequences of the presence of the extrinsic noise in the system.
A first quantity affected by this new source of noise is certainly the average of T. As the level of extrinsic noise is increased, the profile as a function of T T becomes smoother with a less pronounced threshold, while the regions farther away from the threshold are not heavily affected see Fig. Indeed, the effects of the extrinsic noise are stronger in the vicinity of the threshold where the system is ultrasensitive.
Therein, for a fixed T T , small changes in the number of total sequestrants S T can make the system transit form the repressed to the unrepressed regime, bearing significant consequences on the shape of the probability distribution. Indeed, stochastically sampling systems above and below the threshold can result in a bimodal distribution. It corresponds to the aggregated contribution of the repressed systems, i. The second peak is broader and corresponds to the superposition of the unrepressed systems, each of them with its own mean. We remark that the underlying deterministic system is not bistable and that the observed bimodality is due to the stochastic sampling of the repressed and unrepressed regimes across the threshold.
This scenario shows how the threshold feature of the system effectively filters the variability introduced by the extrinsic noise, concentrating the contributions of all the systems below threshold. At this point, one may wonder whether any extrinsic noise would have the same effect on target distributions in any parameter range. The answer is no: bimodal distributions are present only close to the threshold and are favoured by steep thresholds small K d , which allow sampling between the unrepressed and repressed regimes see Fig.
Furthermore, not all distributions of extrinsic noise may induce bimodals for the target. To this aim, the extrinsic noise is required to have a peaked distribution, sufficiently broad to sample both below and above threshold. This becomes clear by examining the Gaussian case with small variances see Fig. In the latter case, the threshold behaviour concentrates the contribution of the systems below threshold, eventually giving rise to a repressed peak, but no mechanisms could induce the appearance of an unrepressed peak see Fig.
S1 in the SI. In view of these results, the combination of the threshold-like response produced by the titrative interaction and a suitable extrinsic noise on the total amount of one of the species can be considered a general mechanism to achieve bimodality. This result is also interesting from a pure biological point of view. In biological systems, bimodal distributions of gene expression are particularly interesting as they may indicate the presence of two distinct physiological states.
At the level of a cell population, this effect would therefore show heterogeneity in different cell gene expression, as observed in How this is normally achieved remains unknown but the mechanism here discussed represents a minimal way to obtain bimodal gene expression distributions in systems based on molecular sequestration and subject to extrinsic noise.
In this case, the system has time to approximately reach a steady state before the amount of sequestrant changes considerably This timescale separation is thought to be present, for instance, in the mechanism of postranscriptional regulation by microRNAs short non-coding segments of RNA The total number of sequestrant molecules and the one of target ones then fluctuates on a slower scale than the sequestration dynamics note that if one focusses on the complex dynamics without explicitly modelling transcription and degradation, the nature of the fluctuations on the number of the sequestrant molecules becomes extrinsic.
We consider here a system composed of two molecular species T 1 and T 2 competing for binding to a third molecule S. The reaction network that defines the minimal model is described by eqs 14 and T 1 can be seen to act as a sponge for the common resource S , preventing the binding with the competing species T 2 and vice versa. Also in this case, we assume the total amount of each species to be constant so that the following conservation laws hold:. These conservation laws limit the number of independent variables from 5 to 2. In particular, if the total number of molecules of a species is larger than the one of its titrant, there will always be some free molecule of that species even if all the available titrant molecules are bound to it so that its minimal value is greater than 0.
When considering two targets, the range of values of free molecules of one target e. In the stochastic description, choosing T 1 and T 2 as independent variables, the model is described by eq. Range of allowed values for sequestrant and targets. The table shows the allowed maximum and minimum value for each species, only based on their relative total amounts. Two targets system. The threshold behaviour of the means and the correlation drop are steeper for lower values of K d.
K 1 d and K 2 d are always equal and assume the values: 5. Once again, detailed balance holds and therefore it is possible to derive the equilibrium solution. As presented in the Methods section, the relations obtained imposing detailed balance, together with the conservation laws, can be used to recursively derive the solution of the master equation, eq. As shown in Fig. The theoretical threshold is again located close to the equimolarity point, i.
The profiles of both averages are ultrasensitive around the threshold and the strength of the ultrasensitivity is controlled by the dissociation constants see Fig. Small values of the dissociation constants mean steeper threshold responses and higher ultrasensitivity. The common interaction between the targets and the sequestrant effectively correlates them see e. When several molecules of a target are bound to the sequestrant, the propensity of binding for molecules of the second target is reduced.
Additionally, when the total amount of molecules of the two targets globally exceeds the sequestrant, having a high number of one target molecules bound implies having fewer molecules of the other target in a complex. From this, it naturally follows that the two targets are negatively correlated via competition. In a later section we will characterise the interaction of the two targets by means of mutual information 30 as done in e. In view of the applications of the present framework to biochemical systems, we focus on the dependence of the correlation on the targets abundances which are simpler to control in biological systems than the targets affinity for the sequestrant.
Nonetheless, we report in the SI a detailed discussion of the dependence of the correlation on the dissociations constants. In Fig. For low dissociation constants, the Pearson coefficient displays a sigmoidal profile red curve, Fig. The shape of the correlation as a function of the total number of one of the targets is affected in a different way by changes of the dissociation constants of the two targets.
Keeping T 2 T fixed, the steepness of the sigmoidal profile as a function of T 1 T is mainly governed by the value of the dissociation constant of target 1, K 1 d , and increases with the decrease of the dissociation constant. Instead, K 2 d , the dissociation constant of target 2, whose abundance is not changed in the plot, affects the minimal value that the correlation asymptotically reaches as T 1 T becomes large see Fig. S3 in the SI. A lower dissociation constant K 2 d corresponds to a lower value of the minimal correlation stronger negative correlation.
However, this smoothens the correlation drop around the threshold and slightly influences its location. No global minimum is found and correlation is stronger more negative as the number of target molecules increases. To investigate the effects of external fluctuations, as done for the case of a single target species, we study the behaviour of the two targets system when the number of total sequestrant molecules is allowed to fluctuate between different realisations of the system.
The probability distribution of S T is again assumed to be a discretised Gaussian and the full probability distribution of the system is obtained as a weighed superposition by implementing the law of total probability. Having full analytical control on the system, the behaviour of the correlation in presence of extrinsic noise can be straightforwardly investigated and precisely quantified.
As a result, we observe that fluctuations in the amount of the sequestrant positively correlate the competing targets. This can be understood by considering the fact that the two target species interact with the same pool of sequestrant molecules. Then, fluctuations in the amount of the sequestrant S T affect in the same way the two target species in each realisation of the extrinsic noise.
For instance, if by an extrinsic fluctuation the amount of sequestrant is lower than the average, both target species will have a higher chance of being free from the sequestrant. This effectively induces a positive correlation between the targets, which can counterbalance the negative one due to competition and discussed in the previous section. The negative interference of these two opposite sources of correlation can result in basically uncorrelated systems. However, these two conflicting sources of correlation depend differently on the rates and the abundances.
The positive correlation is felt the most in the proximity of the threshold. There, most of the sequestrant is bound and neither of the targets have many free molecules. Then, a change say a decrease in the sequestrant abundance directly reflects in a change increase in the number of free molecules of both targets. This is no longer the case when there is an excess of one target e. When the sequestrant largely outnumbers the targets, changes in its abundance will have little impact on the targets, which will mostly be bound to the sequestrant anyway.
From the combination of these effects, correlation takes a non-trivial profile as a function of the abundances and can be positive for sizeable regions of the parameters space. For a more quantitative discussion, we first focus on the correlation profile as a function of the total abundance of target 1, T 1 T , keeping T 2 T fixed.
When the level of extrinsic noise is increased, the positive correlation induced by the sequestrant fluctuations can counterbalance competition, leading to a correlation close to zero. A further increase of extrinsic noise eventually induces a positive correlation between the targets. Because of the different dependence of the sources of correlation on the system features, the resulting profile displays a correlation maximum in the vicinity of the threshold, where the system is more sensitive to fluctuations of the sequestrant amount. This can also be observed by analysing the two-dimensional varying both target abundances profile of correlation shown in Fig.
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