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# Mechanical bounds to transcriptional noise

Contributed by Herbert Levine, October 16, 2016 (sent for review August 3, 2016; reviewed by Erez Braun and Aaron R. Dinner)

## Significance

A complete understanding of transcription, which is the first process in the central dogma of biology, is of paramount importance for understand the living world. Many genes transcribe in a significantly noisy or bursty manner, and the understanding of this phenomenon has been the subject of much experimental and theoretical research. In this article, we present work that introduces mechanical feedback as the source of bursting through an intrinsically ordered stochastic process. The resulting statistical properties of gene expression contain a number of previously unexplained universal features of the central dogma.

## Abstract

Over the past several decades it has been increasingly recognized that stochastic processes play a central role in transcription. Although many stochastic effects have been explained, the source of transcriptional bursting (one of the most well-known sources of stochasticity) has continued to evade understanding. Recent results have pointed to mechanical feedback as the source of transcriptional bursting, but a reconciliation of this perspective with preexisting views of transcriptional regulation is lacking. In this article, we present a simple phenomenological model that is able to incorporate the traditional view of gene expression within a framework with mechanical limits to transcription. By introducing a simple competition between mechanical arrest and relaxation copy number probability distributions collapse onto a shared universal curve under shifting and rescaling and a lower limit of intrinsic noise for any mean expression level is found.

The ability to watch biological phenomena play out at the single-molecule level has revealed a rich and nuanced view of the central dogma of biology. From the single-molecule vantage it has become clear that random forces and events play a key role in transcription (1), although how important expression noise is for crucial biological functions is a matter of debate. The identification of transcriptional bursting, in which genes undergo periods of paused activity even in fully induced environments in both prokaryotes and eukaryotes (2, 3), has been one of the most notable examples of this new perspective. Bursting has also figured prominently in the discussion concerning universal properties of transcriptional noise (4). In particular, a number of recent experimental results have found a link between the rate and randomness of mRNA production whereby highly expressed genes have increased noise associated with production (4). This result transcends specific organisms or genes and may be explained if expression inevitably exhibits bursting. Other work, however, has argued that under some conditions there are nonuniversal gene-specific relationships between the rate and randomness of mRNA production (5); these results are more consistent with the pure model of transcription regulated by the binding of specific regulatory proteins to the promoter regions, in which high expression is not necessarily associated with high noise.

What is needed is a framework that is able to accommodate the traditional “promoter architecture” view of transcription while at the same time capturing recently observed universal aspects of bursting. To accomplish this, we start from the “twin-supercoiling domain” (6) model of transcription wherein the helical nature of DNA combined with topological obstructions leads to the accumulation of mechanical strain in DNA during transcription. This strain can result in arrested gene expression. Specific biological machinery (topoisomerases) must relieve the strain created by transcription through physical, topological manipulation of the DNA for gene expression to continue. A recent study has shed further light on these mechanical aspects of transcription covering both the physical range and speed at which RNA polymerase (RNAP) can operate (7). Additionally the self-induced arresting and topoisomerase-mediated recovery of RNAP during transcription in bacteria (8) has been observed in real time, highlighting the intrinsic role of supercoiling and mechanics in gene expression at the single-transcript level.

Such a mechanically based regulatory scheme acts not independently of, but instead underneath, well-known biological regulatory agents, calling for an extension of the standard view of gene regulation. We believe that this perspective is crucial because mechanical silencing requires transcription so that in systems with mechanical stalling the rate of mechanical arrest is fundamentally tied to the rate of mRNA production.

## Model

A simple way to incorporate the points outlined above is the introduction of strain-induced expression arrest. Transcription induces the buildup of stress, which when it reaches a threshold, causes arrest. The random arrival of topoisomerase is assumed to completely relieve the accumulated stress. If this does not happen in a timely fashion, the stress reaches the threshold and transcription halts until the next random relaxation event. Specifically, we propose the following stochastic model. Each mRNA transcript is made with rate

## Universal Properties

The intrinsically ordered nature of the outlined model results in universal statistical properties of mRNA levels. The probability density functions (PDFs) for mRNA copy number collapse onto the same universal curve with zero mean and standard deviation (SD) of 1 for any model-specific parameters

The existence of a universal PDF

Thus, by using moments

## Master Equation Formulation

The most natural mathematical framework in which to incorporate all of the elements described above is the master equation through which stationary expressions for the mean and variance of mRNA levels can all be analytically derived (*Mathematical Details*). As a transcript is made both the number of transcripts present,

By setting

The second moment of *Mathematical Details*. Analytical expressions were validated with numerical results (Fig. S2). One simple limit of our model is where transcription occurs very quickly compared with the other time scales so that we have a simple process of making a burst of

The relationship between mean and noise levels as a function of promoter strength *A* with the red circles showing the linear dependence in the large *A*), the curve in this scaling limit takes the form

This result, which incorporates the above large

To see how large-scale transcriptional data might appear when different genes are characterized by a spectrum of rates as might exist in real organisms, we have taken the analytical expressions for the mean mRNA levels and Fano factors and generated theoretical data clouds by varying transcriptional production and decay rates *B*. A change in the mechanical barrier through the cutoff value

## Repressor Effects

Most genes are subject to regulation by proteins that couple to DNA and affect RNAP recruitment. For concreteness, we focus on the case of a repressor and simply posit that the production of mRNA will be paused while the gene is repressed. We introduce a hypothetical repressor with binding/unbinding rates

Using the same master equation approach as in the unregulated case we can derive analytical results for the stationary mRNA mean level (*Model with a Repressor: Generating Function Approach* with numerical comparisons in Fig. S3)

In the limit where the mRNA production rate

This formula is very instructive. The Fano factor is composed of two contributions. The first one is that due to the mechanical relaxation alone. The second one is exactly the Fano factor for the pure regulatory dynamics, with

We can again explore all possible noise/mean relationships for a multitude of hypothetical genes faced with mechanical limits and make a direct comparison with previously published universal gene expression noise/mean data (4). The theoretical data points in Fig. 4 represent mean/noise values for mRNA being produced and decaying for a variety of rates *b*, which are subject to mechanical limits, to the black points of Fig. 4*a* that are not. The introduction of a mechanical limit to transcription naturally generates previously unexplained bounds to the mean and intrinsic noise found in experimental data (4, 11).

In addition to changing expression levels by varying **7**, where the Fano factor expression has a nearly identical form to the Fano factor for a gene with only repression except that where we would normally find the transcriptional rate

## Burst Properties

Beyond noise/mean statistical data, wait-time distributions for periods of activity and inactivity, as well as the burst-size distributions for the number of mRNAs made in-between arrests, can shed light on the nature of the arrest process. We operationally defined bursts as periods without mRNA production lasting more than a time *A*) but exponential long-time behavior as well as a simple burst-size distribution where the maximum number of transcripts *B*). However, when the transcription rate is more comparable to relaxation, the peaks of the wait-time distributions become broadened and the distributions show strong exponential behavior (dashed curves in Fig. 5*A*). Additionally, for slowly arresting genes, a much broader burst-size distribution, where often many more than *B*), showing a strong resemblance to a geometric distribution.

We calculate the wait-time distributions for the probability of taking time

We need only the transitions between the states and have no need to incorporate RNA production/degradation; thus we are interested in the master equation dynamics of the states by themselves:

Taking the Laplace transform yields

Adding these all together yields

Thus

The closed-form expression for *A* contains plots of

## Protein Statistics

It is important to examine the effects of mechanical arrest in transcription on protein levels. To do this we incorporated an additional single-step translational process into our stochastic simulations where a protein is made with rate proportional to the mRNA content as

The resulting probability functions for protein levels using stochastic simulations are shown in Fig. 6 where the universal nature of the transcriptional arrest is manifested in the protein statistics. Although the resulting universal curve is not identical to the one found in ref. 9, there is strong agreement with only moderate deviation at large expression levels. Additional effects such as more complex translational mechanisms, copy number variation, and variable decay likely contribute to the observed disagreement. The most important properties of zero mean and SD of one are found over a wide parameter range.

## Mathematical Details

We are constructing a master equation for the state probabilities

### Relaxed (α = 0 ).

The master equation is given by

### Arrested (α = m c ).

The master equation is given by

### Bulk (1 ≤ α ≤ m c − 1 ).

The master equation is given by

### Total Probability.

Iteratively we can get the probabilities for the bulk as

### Total Mean.

The equation for the total generating function equation is obtained by just adding all of the equations together to find

Using the standard expression for the mean in terms of the generating function we find

It is interesting that

### Variance/Fano Factor.

Using the expression for the first moment we can find the second moment

This gives

So then we are left with figuring out

### Relaxed.

### Arrested.

### Bulk.

### Solution.

This is a bit complicated due to the recursive nature of the mean. First we must write the bulk mean contributions to the total mean in terms of the first state. The inhomogeneous linear recursion relation Eq. **S21** has the solution

Then, by Eq. **S18**,

We can do the sum because, from Eq. **S7**, the

The analytical expressions are compared with numerical simulations in Fig. S2.

### Curve Scaling.

We are especially interested in obtaining the functional form for the Fano factor mean curves when

For

Similarly, for the variance

These expressions become more clear in particular cases such as for

We see that

Finally for

## Discussion

It is worth noting that the full nature of the mechanical arrest and relaxation is not addressed within this article. However, the simple phenomenological model presented here is a first attempt at putting together the necessary ingredients for understanding this more detailed problem and appears to capture several essential features of transcriptional noise. As a simple check we conducted numerical simulations (shown in Fig. S5 as points) for a gene that experiences decreased proclivity for expression as the cutoff *Supporting Information*). Additionally a similar attempt to understand mechanical aspects of transcription has been made (12) where supercoiling-induced transcriptional slowing was numerically shown to increase noise. We expect further theoretical and experimental work on the role mechanical effects play in gene expression to refine, but not significantly change, the perspective presented here.

In addition to the work presented here Brenner et al. (13, 14) have shown other ordered processes (growth and division) can generate universal statistical distributions for protein copy numbers in populations (13) and individual microorganisms (14). The ordered recursive nature of growth and division has shared mathematical roots and structure with the ordered mechanical arrest process presented. Furthermore, the results presented in this work are not inconsistent with additional cellular processes (such as growth and division) but do provide a unification of various disparate phenomena that have not been previously joined. We have conducted simple simulations demonstrating that the universal nature of the mechanical arrest process is not destroyed by introducing additional processes. Consequently, we believe that the incorporation of random growth and division could correct the slight disagreement between our distributions and the experimental ones. These efforts are left for future work.

Finally, the theoretical discussion within this article has not centered on any particular organism. Although how the work here fits into a larger panoply of DNA structure is certainly a matter of current interest and research, we believe that the presented framework is capable of capturing the same phenomena in many organisms. This may offer an explanation for the noise/mean relationship observed in both bacteria and higher organisms (4). Although the most direct evidence for mechanical arresting exists in bacterial systems, the first discovery of topoisomerases and transcription-induced supercoiling was done in eukaryotes (15). Additionally in yeast it has been demonstrated that transcription can occur without topoisomerases (15) possibly due to reduced chromosome organization (16). Thus, mechanical arresting provides a mechanism through which previously unexplained mean and noise bounds in transcriptional data across many organisms (4) can be reconciled with the prevailing promoter architecture view of transcription.

Additionally the robust appearance of universal properties highlights the importance of the ordered nature of mechanical arrest and illustrates the conceptual and mathematical differences between the model proposed in this article and previously considered models of transcriptional bursting.

## Conclusion

In conclusion the results of this article are beginning steps in combining the traditional view of transcriptional regulation with recent discoveries concerning the role of stochasticity and mechanics. The reconciliation of these two perspectives, even at the simple level presented here, is key to resolving outstanding puzzles in gene expression and allowing for a more complete view of transcription to emerge. It is promising that even at the simple level presented here highly nontrivial universal properties emerge.

## Model with a Repressor: Generating Function Approach

We now repeat the same formulation but incorporate a suppressed state by constructing a generating function for each state that is unsuppressed

### Equations.

#### Relaxed.

The master equations are given by

#### Arrested.

The master equations are given by

#### Bulk.

The master equations are given by

#### Together.

Then

#### Mean.

Finding the probability

The result for the unsuppressed, arrested state

We can now tackle the

Similarly, for the bulk states,

For the relaxed state we have master equations

Iterating Eq. **S41** together with Eq. **S46**, we have

### Fano Factor.

We now turn to the calculation of

#### Arrested.

Starting from Eq. **S32**, setting the time derivatives to zero and taking

This gives

#### Bulk.

The generating equations are

#### Relaxed.

The generating equations are

We know **S49**, **S52**, and **S55** and find

Then, using **S53**, we have**S50****S41**).

The analytical expressions are compared with numerical simulations below. All results are for a gene with fixed

### Limits.

In the fast transcriptional limit for a gene with repression the

In the limit of

This illustrates how noise much greater than the lower bound set by the mechanical limit can exist for various repressor kinetics.

In Fig. S4 we show curves demonstrating mean/Fano relationships for varying binding kinetics

These results are used to generate wait-time probability distributions in Fig. 4*A* of the main text.

## Model Refinements

### Slowed Transcription.

Within the basic phenomenological model outlined here there are a number of simplifications that we have used that are worth addressing, most notably the nature of the mechanical arrest and relaxation processes. In our simplified model the production of any given mRNA adds the same mechanical strain to the system and each occurs with the same random rate *r* is the baseline rate and

## Acknowledgments

We thank Samuel Skinner and Ido Golding for helpful discussions. This work was supported by the National Science Foundation Center for Theoretical Biological Physics (Grant NSF PHY-1308264). H.L. was also supported by the Cancer Prevention and Research Institute of Texas Scholar program of the State of Texas. D.A.K. was supported by the Israel Science Foundation (Grant 376/12).

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: herbert.levine{at}rice.edu or s.a.sevier{at}rice.edu.

Author contributions: S.A.S., D.A.K., and H.L. designed research, performed research, and wrote the paper.

Reviewers: E.B., Technion–Israel Institute of Technology; and A.R.D., The University of Chicago.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1612651113/-/DCSupplemental.

## References

- ↵
- ↵
- ↵
- ↵.
- Sanchez A ,
- Golding I

- ↵.
- Jones DL ,
- Brewster RC ,
- Phillips R

- ↵.
- Liu LF ,
- Wang JC

- ↵.
- Ma J ,
- Bai L ,
- Wang MD

- ↵
- ↵
- ↵
- ↵
- ↵.
- Bohrer CH ,
- Roberts E

- ↵.
- Brenner N , et al.

- ↵.
- Brenner N , et al.

- ↵
- ↵

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