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Conceived and designed the experiments: KJ WO MRB. Performed the experiments: KJ JML WO MRB. Analyzed the data: KJ JML WO LS MRB. Wrote the paper: KJ JML WO LS MRB.

The authors have declared that no competing interests exist.

The creation of protein from DNA is a dynamic process consisting of numerous reactions, such as transcription, translation and protein folding. Each of these reactions is further comprised of hundreds or thousands of sub-steps that must be completed before a protein is fully mature. Consequently, the time it takes to create a single protein depends on the number of steps in the reaction chain and the nature of each step. One way to account for these reactions in models of gene regulatory networks is to incorporate dynamical delay. However, the stochastic nature of the reactions necessary to produce protein leads to a waiting time that is randomly distributed. Here, we use queueing theory to examine the effects of such distributed delay on the propagation of information through transcriptionally regulated genetic networks. In an analytically tractable model we find that increasing the randomness in protein production delay can increase signaling speed in transcriptional networks. The effect is confirmed in stochastic simulations, and we demonstrate its impact in several common transcriptional motifs. In particular, we show that in feedforward loops signaling time and magnitude are significantly affected by distributed delay. In addition, delay has previously been shown to cause stable oscillations in circuits with negative feedback. We show that the period and the amplitude of the oscillations monotonically decrease as the variability of the delay time increases.

Delay in gene regulatory networks often arises from the numerous sequential reactions necessary to create fully functional protein from DNA. While the molecular mechanisms behind protein production and maturation are known, it is still unknown to what extent the resulting delay affects signaling in transcriptional networks. In contrast to previous studies that have examined the consequences of fixed delay in gene networks, here we investigate how the variability of the delay time influences the resulting dynamics. The exact distribution of “transcriptional delay” is still unknown, and most likely greatly depends on both intrinsic and extrinsic factors. Nevertheless, we are able to deduce specific effects of distributed delay on transcriptional signaling that are independent of the underlying distribution. We find that the time it takes for a gene encoding a transcription factor to signal its downstream target decreases as the delay variability increases. We use queueing theory to derive a simple relationship describing this result, and use stochastic simulations to confirm it. The consequences of distributed delay for several common transcriptional motifs are also discussed.

Gene regulation forms a basis for cellular decision-making processes and transcriptional signaling is one way in which cells can modulate gene expression patterns

The majority of models, however, are systems of nonlinear ordinary differential equations (ODEs). Yet, because of the complexity of protein production, ODE models of transcriptional networks are at best heuristic reductions of the true system, and often fail to capture many aspects of network dynamics. Many ignored reactions, like oligomerization of transcription factors or enzyme-substrate binding, occur at much faster timescales than reactions such as transcription and degradation of proteins. Reduced models are frequently obtained by eliminating these fast reactions

Delay differential equations (DDEs) have been used as an alternative to ODE models to address this problem. In protein production, one can think of delay as resulting from the sequential assembly of first mRNA and then protein

Protein production delay times are difficult to measure in live cells, though recent work has shown that the time it takes for transcription to occur in yeast can be on the order of minutes and is highly variable

In this study, we examine the consequences of randomly distributed delay on simple gene regulatory networks: We assume that the delay time for protein production,

Equation (1) only holds in the limit of large protein numbers

Queueing theory has recently been used to understand the behavior of genetic networks

The transcription of genetic material into mRNA and its subsequent translation into protein involves potentially hundreds or thousands of biochemical reactions. Hence, detailed models of these processes are prohibitively complex. When simulating genetic circuits it is frequently assumed that gene expression instantaneously results in fully formed proteins. However, each step in the chain of reactions leading from transcription initiation to a folded protein takes time (

(A) Numerous reactions must occur between the time that transcription starts and when the resulting protein molecule is fully formed and mature. Though we call this phenomenon “transcriptional” delay, there are many reactions after transcription (such as translation) which contribute to the overall delay. (B) The creation of multiple proteins can be thought of as a queueing process. Nascent proteins enter the queue (an input event) and emerge fully matured (an output event) some time later depending on the distribution of delay times. Because the delay is random, it is possible that the order of proteins entering the queue is not preserved upon exit. (C) In a transcriptionally regulated signaling process the time it takes for changes in the expression of gene 1 to propagate to gene 2 depends on both the distribution of delay times,

In one recent study, Bel

However, it is possible, and perhaps likely, that the limiting distributions described by Bel

If the biochemical reaction pathway that leads to functional protein is known and relatively simple, direct stochastic simulation of every step in the network is preferable to simulation based on scheme (2). From the point of view of multi-scale modeling, however, paradigm (2) is useful when the biochemical reaction network is either extremely complex or poorly mapped, since one needs to know only the statistical properties of

In the setting of scheme (2), first assume that

In our model, the order in which initiation events enter a queue is not necessarily preserved. As

One purpose of transcription factors is to propagate signals to downstream target genes. Determining the dynamics and stochasticity of these signaling cascades is of both theoretical and experimental interest

We now ask: If

We first examine reaction (3a). Assume that at time

The exit process,

Inactivation (or activation) of gene 2 occurs when enough protein

The probability density function of

Consequently, the mean and variance of the time it takes for the original signal to propagate to the downstream gene can be written as:

To gain insight into the behaviors of Eqs. (6) and (7), we first examine a representative, analytically tractable example. Assume that the delay time can take on

(A) For the simplified symmetric distribution where the delay takes values

It follows that for larger delay variability, the mean signaling time

The bottom row of

We therefore expect that for each fixed threshold

In sum, mean signaling times decrease as delay variability increases (with fixed mean delay). This effect is most significant for small to moderate thresholds. We note that the decrease in mean signaling time phenomenon depends on a sufficient number of proteins entering the queue. If transcription is only active long enough for less than

Using the above results, we now examine more complicated transcriptional signaling networks. In particular, we turn to two common feedforward loops - the type 1 coherent and the type 1 incoherent feedforward loops (FFL)

Each pathway in the networks has an associated signaling threshold (

To examine the effect of distributed delay on these networks we assume that at

The signaling time between any two nodes

In contrast to the coherent FFL, the dynamics of the pulse generating incoherent FFL are less trivial. Since the repressor (

(A)

To see this, write

These observations can also be extended to networks with recurrent architectures. For instance, consider the transcriptional delayed negative feedback circuit

As a result during each oscillation the gene is turned on until its own signal reaches itself, at which time the gene is turned off

Shown are the analytically predicted (solid lines) and numerically obtained (symbols with standard deviation error bars) mean peak heights of the negative feedback oscillator with Hill coefficients of

We can use our theory to predict the change in the peak height of the oscillator as a function of

However, due to degradation, Eq. (14) overestimates the correction to the peak height. Due to exponential degradation, only a fraction

The existence of delay in the production of protein has been known of for some time. For many systems its presence does not seriously impact performance. For example, the existence of fixed points in simple downstream regulatory networks without feedback is unaffected by delay. Delay is important if the timing of signal propagation impacts the function of the network. Delay can also change a network's dynamics. In networks with feedback, for instance, delay can result in bifurcations that are not present in the corresponding non-delayed system. The delayed negative feedback oscillator is a prime example

The intrinsic stochasticity of the reactions that create mature protein make some variation in delay time inevitable. However, we do not yet know the exact nature of this variability or the functional form of the probability density function

We focused on the transient dynamics of

The complexity of biochemical reaction networks suggests the use of networks of queues

One further complication occurs if the burstiness of the promoter is large

One can think of

We first derive the signaling time distributions for a single gene that is modeled by an

Thus

(transient distributions; see

Let

Define the rescaled departure process

Computing the expectation of

We now show that if

Suppose that

If

Computing the expectation of

Suppose that the rate function of the input process is constant and equal to

Let

For

The signaling time

We compute

Substituting for

Using (17) and (19b), we have

Finally, we express (21) using gamma functions:

We now examine the asymptotics of

Expanding

Using the Stirling approximation

In particular, for

In this case the correction to the deterministic limit is of order

We obtain linear large

(A) PDFs for the signaling time using the delay distribution

Suppose that the rate function of the input process is constant and equal to

The CDF of

Expanding

Note that the corrections to the first terms in the expansions are exponentially small in

Using (19a), we have

Consider a network of two

Variances of signaling times propagate additively through linear chains of genes in which each gene up-regulates the next. Let

Analogously, let

This argument extends inductively to directed pathways in which the product of each gene activates the subsequent gene in the sequence.

Suppose that

For a repressor switch, the

If the midpoint

We now examine the ability of gene

increases as

does not depend on

decreases as

Intuitively, this is due to the fact that the order in which proteins enter the queue is not necessarily the same as the order in which they exit. Consider again

We conjecture that this trichotomy holds in general if

Gillespie's stochastic simulation algorithm generates an exact stochastic realization for a system of

The idea behind extending Gillespie's SSA to model distributed delay is that if a reaction is to be delayed by some amount of time then we temporarily store this reaction along with the time at which the event will occur and we only apply this reaction at the given time. We used a version of the algorithm equivalent to those described in

We wish to thank R. E. Lee DeVille for helpful discussions.