BACKGROUND
in prokaryotes, both transcription and translation are stochastic, multi-stepped processes that involve many components and chemical interactions.
several events in transcription and in translation  <cit>  are probabilistic in nature, and their kinetics are sequence dependent.
one example is sequence-dependent transcriptional pausing  <cit> .
when they occur, these events can affect the degree of fluctuations of rna and protein levels.
since noise in gene expression affects cellular phenotype, sequence dependent noise sources are subject to selection  <cit>  and are thus evolvable  <cit> .
recent evidence suggests that these noise sources may be key for bacterial adaptability in unpredictable or fluctuating environmental conditions  <cit> .
to better understand the evolvability of bacteria, it is important to understand how fluctuations in rna levels propagate to protein levels.
transcription and translation are coupled in prokaryotes, in that translation can initiate after the formation of the ribosome binding site region of the rna, which occurs during the initial stages of transcription elongation.
the extent to which sequence-dependent events in transcription elongation affect the noise in rna, and consequently protein levels is largely unknown.
due to this, it is also not yet well understood how phenotypic diversity is regulated in monoclonal bacterial populations.
two recent experiments have given a preliminary glimpse at the dynamics of production of individual proteins  <cit>  and rna molecules  <cit>  in vivo in bacteria.
however, as of yet, there is no experimental setting to simultaneously observe the production of both rna and proteins at the molecular level.
further, in the aforementioned experiments  <cit> , the rate of gene expression was kept very weak, as otherwise the number of molecules would not be easily quantifiable.
this implies that they cannot be used to study the effects of events such as the promoter open complex formation  <cit> .
the present shortcomings of these techniques enhance the need for realistic models of gene expression in prokaryotes.
several measurements have shed light on the dynamics of transcription and translation elongation  <cit> , and revealed the occurrence of several stochastic events during these processes, such as transcriptional pauses  <cit> .
the kinetics of rna and protein degradation are also better known  <cit> .
these measurements allowed the recent development of realistic kinetic models of transcription at the nucleotide level  <cit>  and translation at the codon level  <cit> .
these models were shown to match the measurements of rna production at the molecule level  <cit>  and of translation elongation dynamics at the codon level  <cit> .
in this regard, it was shown that measurements of sequence dependent translation rates of synonymous codons could be modeled with neither deterministic nor uniform stochastic models  <cit> , thus the need for models with explicit translation elongation.
similarly, transcription elongation also needs to be modeled explicitly to accurately capture the fluctuations in rna levels for fast transcription initiation rates  <cit> .
here, we propose a model of transcription and translation at the nucleotide and codon level for escherichia coli.
the model of transcription is the same as in  <cit> , and includes the promoter occupancy time, transcriptional pausing, arrests, editing, premature termination, pyrophosphorolysis, and accounts for the rnap footprint in the dna template.
the model of translation at the codon level proposed here is based on the codon-dependent translation model proposed in  <cit> , which includes translation initiation, codon-specific translation rates and the stepwise translation elongation and activation.
the model also accounts for the ribosome's footprint in the rna template as well as the occupancy time of the ribosome binding site.
here, beside these features, we further include the processes of back-translocation, drop-off, and trans-translation.
finally, we include protein folding and activation, as well as degradation, modeled as first-order processes, so as to study fluctuations in the protein levels.
the dynamics of the model follow the delayed stochastic simulation algorithm  <cit>  and is simulated by a modified version of sgnsim  <cit> .
while the most relevant innovation is the coupling between realistic stochastic models of transcription and translation at the nucleotide and codon levels, which allows the study of previously unaddressed aspects of the dynamics of gene expression in prokaryotes, this introduces a level of complexity that required simulation capabilities that sgnsim did not possess.
namely, the simulator is required to create and destroy compartments at run time within the reaction vessel, where a separate set of reactions can occur.
we start by validating the dynamics of translation elongation in the model.
next, using realistic parameter values extracted from measurements, we address the following questions: how different are the distributions of time intervals between translation initiation events and between translation completion events, i.e., how stochastic is translation elongation?
to what extent do fluctuations in temporal rna levels propagate to temporal protein levels, and what physical parameters control this propagation of noise between the two?
finally, we investigate whether transcriptional pauses have a significant effect on the dynamics of protein levels.
RESULTS
dynamics of transcript production
given the number of chemical reactions per nucleotide in the model and that one gene can have thousands of nucleotides, the dynamics are considerably complex.
to illustrate this, we show examples of the kinetics of multiple rnaps on a dna strand within a short time interval, and the dynamics of multiple ribosomes on one of the rna strands as it is transcribed.
parameter values were obtained from measurements in e. coli for lacz , since the dynamics of transcription and translation have been extensively studied for this gene.
lacz has  <dig> nucleotides and its transcription is controlled by the lac operon.
in this simulation, transcription is not repressed.
thus, provided that the promoter is available for transcription, the expected time for a transcription event to start is approximately  <dig>  s, given the value of the rate constant of reaction  in table  <dig> and that there are  <dig> rnap molecules available in the system  <cit> .
the promoter open complex formation step, with a mean duration of  <dig> s  <cit>  and a standard deviation of  <dig> s  <cit>  is the major limiting factor of transcription events in these conditions.
chemical reactions, rate constants , and time delays  used to model transcription initiation, elongation, and termination.
parameter values were obtained from measurements in e. coli, mainly for lacz.
references are reported in the column ref.
dynamics of production of proteins
comparing the dynamics of the model of translation with measurements
recently, the real-time expression of a lac promoter was directly monitored in e. coli with single-protein resolution  <cit> .
the proteins were found to be produced in bursts , with the distribution of intervals between bursts fitting an exponential distribution, while the number of proteins per burst followed a geometric distribution  <cit> .
these distributions were measured for a gene that was kept strongly repressed and for which the ribosome binding site  was engineered so that translation was also very weak  <cit> .
under these conditions, our model reproduces these dynamics .
nevertheless, we note that it is possible to match these measurements with a simpler model than the one proposed here, where transcription and translation are modeled as single step events  <cit> .
we next compare the kinetics of translation in our model with measurements of the translation elongation speed in three engineered e. coli strains designed to enhance queue formation and traffic in translation  <cit> .
each strain contains a different mutant of lacz.
the pmas <dig> strain corresponds to the wild-type lacz.
the other two sequences differ in that a region of slow-to-translate codons was inserted .
the speed of protein chain elongation was measured by subjecting the cells to a pulse of radioactive methionines, and then measuring the level of radioactivity in cells of each population, every  <dig> s after the pulse.
each strand contained  <dig> methionines, spread out unevenly on the dna sequence, causing the incorporation curve to be non-linear.
given that they differ in the nucleotide sequence, it was hypothesized that the translation elongation speed of the three strands would differ, as the speed of incorporation of an amino acid depends on which synonymous codon is coding for it  <cit> .
the cells where translation is faster will thus be expected to have higher levels of radioactivity in the translated proteins, as more labeled amino acids have been incorporated in a fixed time interval.
if the translation speeds of the three strands were identical, they would exhibit identical levels of radioactivity at the same point in time.
to model this, we simulate the transcription and translation processes of the three sequences  <cit> .
we model the incorporation of radioactive methionines at the same locations as in these sequences.
the three model strands differ only in sequence, as in the measurements.
during the simulations, we measure the number of incorporated radioactive methionines at the same points in time as in the experiment.
results of our simulations and of the measurements  <cit>  are shown in figure  <dig>  showing good agreement between model and measurements.
propagation of fluctuations in rna levels to protein levels
we simulate the model for varying effective rates of transcription initiation .
this rate is determined by the basal rate of transcription initiation , which sets the binding affinity of the rnap to the transcription start site, and by the strength of repression of transcription.
thus, to vary keff, we vary the number of repressor molecules present in the system.
three sets of simulations are performed, differing in rate of translation initiation .
this rate is one of the kinetic parameters of the model, thus can be changed directly, and not by indirect means as keff.
in e. coli genes, this rate is believed to be determined by the rbs sequence  <cit> .
mrna and protein degradation rates are set so that the mrna and protein mean levels are identical for all cases, allowing us to study how the level of noise in mrna and protein levels changes.
for each set of values of keff and ktr we perform  <dig> independent simulations.
depending on these rates, the mean time to reach steady state differs.
each case is simulated for long enough to reach steady state and for an additional  <dig>  <dig> s after that.
the time series of the  <dig> simulations for each set of parameter values is concatenated into one time series, from which the noise is quantified by the square of the coefficient of variation, cv <dig>   <cit> .
this number of long simulations is necessary to properly sample the system due to the stochasticity of the underlying processes.
in figure  <dig>  we first show the cv <dig> of mrna time series for varying keff.
noise decreases as keff increases due to the promoter open complex formation step  <cit> .
without this event, the distribution of time intervals between transcription initiation events would be exponential, and the cv <dig> would not vary.
however, with this step, if the expected time for an rnap to bind to the free promoter is faster than the duration of the promoter open complex formation, then the distribution of time intervals becomes gaussian-like  <cit> .
no measurements have yet been made to study experimentally the relation between the noise in mrna levels and the corresponding protein levels.
nevertheless, it is possible to create a robust estimate, provided reasonable assumptions on the nature of the underlying processes  <cit> .
our model allows for a direct assessment, and it additionally includes realistic events such as rnap and ribosome traffic, in transcription and translation elongation, which are not included in the aforementioned estimations  <cit> .
figure  <dig> shows the noise  in protein levels, for varying keff and three values of ktr.
the data was obtained from the same simulations used to generate the results in figure  <dig>
in general, we find that increasing keff decreases the noise in protein levels due to the decrease of noise in mrna levels.
increasing ktr increases the noise in protein levels, due to the increased size of the bursts in the protein level  <cit> .
this finding has not yet been experimentally validated by direct means.
an interesting observation from figures  <dig> and  <dig> is that, for keff <  <dig> × 10- <dig> s- <dig>  as keff is increased, the noise in protein levels decreases significantly, while the noise in rna levels does not noticeably change.
this is due to the decrease in mean protein burst size, i.e., the mean number of proteins produced from each rna molecule, as both keff and the degradation rate of rna molecules are varied.
from these results, we conclude that the degree of coupling between transcription and translation is likely to be a key determining factor of the noise in protein levels.
this can be verified by computing the normalized maximum correlation between time-series of protein and mrna levels for each set of parameter values .
comparing figures  <dig> and  <dig>  we see that higher correlation values are obtained for the regime of higher noise in the protein levels.
this implies that the principal source of this noise is the fluctuations in rna levels.
the correlation value is largely determined by the rates of mrna and protein degradation and production.
for example, both increasing the mrna degradation rate and/or decreasing the protein degradation rate increases the time averaging constant of the mrna fluctuations, and thus decreases the correlation between mrna and protein levels.
in general, if the mean mrna and protein levels and kept unchanged by tuning their degradation rates accordingly, the correlation between rna and protein time series can be increased by lowering the mrna production rate and/or increasing the protein production rate.
effects of transcriptional pauses on the fluctuations in protein levels
recent work  <cit>  reported that long transcriptional pauses enhance the noise in mrna levels.
we next investigate to what extent the fluctuations in rna levels caused by long transcriptional pauses propagate to protein levels.
long sequence-dependent pauses  <cit>  in transcription elongation may cause the ribosome to stall in the mrna chain.
this will likely cause subsequent ribosomes to accumulate in the preceding sequence.
when the rnap is spontaneously released from the pause  <cit> , translation of the stalled ribosomes likely resumes but the distribution of intervals between them will differ significantly from what it would have been without the pause event.
consequently, the protein production is likely to become burstier, especially if the long pause site is located near the end of the sequence.
an increase in burstiness ought to increase the noise in protein levels.
to verify this, we perform two simulations.
we introduce a long-pause sequence with mean pause durations of  <dig> s in one case, and  <dig> s in the other .
in both cases, we set the probability that an rnap will pause at that site to 70% .
measuring the protein noise levels, we find that the cv <dig> is ~5% higher for the  <dig> s pause site and ~10% higher for the  <dig> s pause site, in comparison to the same sequence without any sequence specific long-pause site.
these relative differences can be biologically relevant in that such a change may, in some cases, cause the degree of phenotypic diversity of a monoclonal cell population to change.
the effects of several pause sites on the same strain are cumulative, namely, the higher the number of pause sites, the higher the noise in rna levels  <cit> .
combined with the present results, this leads us to the conclusion that the sequence-dependent transcriptional pausing mechanism likely exists to allow a wide variation of both rna and protein noise levels.
CONCLUSIONS
we proposed a new delayed stochastic model of prokaryotic transcription and translation at the single nucleotide and codon level, where the processes of transcription and translation are dynamically coupled in that translation can initiate immediately upon the formation of the ribosome binding site region of the nascent mrna.
simulations of the model's dynamics show that, within realistic parameter values, the protein noise levels are determined, to a great extent, by the fluctuations in the rna levels, rather than from sources in translation, in agreement with indirect measurements  <cit> , as translation elongation was found to be less stochastic than transcription elongation.
specifically, the distributions of intervals between translation initiation and translation completion events only differ significantly if the sequence possesses long sequence-dependent pauses or clusters of slow-to-translate codons.
the sequence dependence of several mechanisms that can act as generators of strong fluctuations in rna levels  <cit> , the propagation of these fluctuations to protein levels, and the ability of fluctuations in protein levels to affect cellular phenotype  <cit> , suggest that these mechanisms may be evolvable.
as a previous study has suggested  <cit> , the translation initiation rate was found to be key in determining the degree of coupling between the fluctuations in rna and protein levels, if one assumes that the degradation rate of the proteins is changed accordingly to maintain their mean level unchanged.
varying this sequence-dependent, and thus, evolvable parameter  <cit>  within realistic ranges gave a widely varying degree of coupling between the fluctuations in rna and protein levels.
it is therefore not necessarily true that noisy production of rna molecules results in noisy protein levels.
interestingly, while decreasing the coupling between transcription and translation by decreasing the rate of translation initiation causes the protein levels to become less noisy, it also takes longer for a change in rna levels to be followed by the protein levels.
this suggests that to be able to change rapidly in response to, e.g., environmental changes, the levels of a protein will be necessarily noisier.
confirming previous studies  <cit> , we found that the distributions of time intervals between transcription initiation and completion events differ significantly and that the faster the rate of transcription initiation events, the more they differ.
this implies that in the regime of fast transcription, both the transcription and translation elongation processes need to be modeled explicitly and coupled, if one is to match the mean and fluctuations in the protein levels at the molecular level.
this is of relevance, since bursts in protein levels may trigger many processes, such as phenotypic differentiation  <cit> .
a final justification for using the model proposed here is the complexity of the process of gene expression in e. coli, and the fact that many events therein may or may not affect the temporal rna and protein levels significantly, depending on their specific sequence-dependent features.
such effects, due to the complexity of the system, are not easily predictable without performing explicit numerical simulations.
the model proposed here includes several features not included in previous models such as a gradual degradation event that can be triggered while the rna is still being transcribed.
as its parameter values were extracted from measurements, it should be useful in the study of several aspects of the dynamics of gene expression in prokaryotes that cannot yet be measured directly and to explore the state space of gene expression dynamics by varying any of the physical variables within realistic ranges.
however, the present model does not yet account for known effects of ribosomes on the dynamics of transcription elongation.
these might need to be included in future developments of the proposed model as recent results  <cit>  suggest that the rate of translation elongation can affect the rate of transcription elongation, due to possible interactions between the ribosome that first binds to the mrna and the rnap transcribing it.
possible effects may include facilitating the release of paused rnap's, which could affect the degree of the contribution of pauses to the noise in rna and thus protein levels.
we do not exclude the possibility that the contrary may occur in specific cases, that is, that the paused state of the rnap may cause pauses in the ribosome translational dynamics, which would amplify the effect of transcriptional pauses on the fluctuations of protein levels.
whether the pause is ubiquitous or due to loop formations in the nascent rna may affect the results of the interaction as well.
provided experimental evidence on the nature and consequences of these interactions, once included in the model, we may be able to test, among other things, whether long transcriptional pauses located in an attenuator system provide an additional layer of control over premature transcription terminations, and thus over rna and protein noise levels.

 in prokaryotes, both transcription and translation are stochastic, multi-stepped processes that involve many components and chemical interactions. several events in transcription and in translation are probabilistic in nature, and their kinetics are sequence dependent. one example is sequence-dependent transcriptional pausing. when they occur, these events can affect the degree of fluctuations of rna and protein levels. since noise in gene expression affects cellular phenotype, sequence dependent noise sources are subject to selection and are thus evolvable. recent evidence suggests that these noise sources may be key for bacterial adaptability in unpredictable or fluctuating environmental conditions. to better understand the evolvability of bacteria, it is important to understand how fluctuations in rna levels propagate to protein levels. transcription and translation are coupled in prokaryotes, in that translation can initiate after the formation of the ribosome binding site region of the rna, which occurs during the initial stages of transcription elongation. the extent to which sequence-dependent events in transcription elongation affect the noise in rna, and consequently protein levels is largely unknown. due to this, it is also not yet well understood how phenotypic diversity is regulated in monoclonal bacterial populations. two recent experiments have given a preliminary glimpse at the dynamics of production of individual proteins and rna molecules in vivo in bacteria. however, as of yet, there is no experimental setting to simultaneously observe the production of both rna and proteins at the molecular level. further, in the aforementioned experiments, the rate of gene expression was kept very weak, as otherwise the number of molecules would not be easily quantifiable. this implies that they cannot be used to study the effects of events such as the promoter open complex formation. the present shortcomings of these techniques enhance the need for realistic models of gene expression in prokaryotes. several measurements have shed light on the dynamics of transcription and translation elongation, and revealed the occurrence of several stochastic events during these processes, such as transcriptional pauses. the kinetics of rna and protein degradation are also better known. these measurements allowed the recent development of realistic kinetic models of transcription at the nucleotide level and translation at the codon level. these models were shown to match the measurements of rna production at the molecule level and of translation elongation dynamics at the codon level. in this regard, it was shown that measurements of sequence dependent translation rates of synonymous codons could be modeled with neither deterministic nor uniform stochastic models, thus the need for models with explicit translation elongation. similarly, transcription elongation also needs to be modeled explicitly to accurately capture the fluctuations in rna levels for fast transcription initiation rates. here, we propose a model of transcription and translation at the nucleotide and codon level for escherichia coli. the model of transcription is the same as in, and includes the promoter occupancy time, transcriptional pausing, arrests, editing, premature termination, pyrophosphorolysis, and accounts for the rnap footprint in the dna template. the model of translation at the codon level proposed here is based on the codon-dependent translation model proposed in, which includes translation initiation, codon-specific translation rates and the stepwise translation elongation and activation. the model also accounts for the ribosome's footprint in the rna template as well as the occupancy time of the ribosome binding site. here, beside these features, we further include the processes of back-translocation, drop-off, and trans-translation. finally, we include protein folding and activation, as well as degradation, modeled as first-order processes, so as to study fluctuations in the protein levels. the dynamics of the model follow the delayed stochastic simulation algorithm and is simulated by a modified version of sgnsim. while the most relevant innovation is the coupling between realistic stochastic models of transcription and translation at the nucleotide and codon levels, which allows the study of previously unaddressed aspects of the dynamics of gene expression in prokaryotes, this introduces a level of complexity that required simulation capabilities that sgnsim did not possess. namely, the simulator is required to create and destroy compartments at run time within the reaction vessel, where a separate set of reactions can occur. we start by validating the dynamics of translation elongation in the model. next, using realistic parameter values extracted from measurements, we address the following questions: how different are the distributions of time intervals between translation initiation events and between translation completion events, i.e., how stochastic is translation elongation? to what extent do fluctuations in temporal rna levels propagate to temporal protein levels, and what physical parameters control this propagation of noise between the two? finally, we investigate whether transcriptional pauses have a significant effect on the dynamics of protein levels.  dynamics of transcript production given the number of chemical reactions per nucleotide in the model and that one gene can have thousands of nucleotides, the dynamics are considerably complex. to illustrate this, we show examples of the kinetics of multiple rnaps on a dna strand within a short time interval, and the dynamics of multiple ribosomes on one of the rna strands as it is transcribed. parameter values were obtained from measurements in e. coli for lacz, since the dynamics of transcription and translation have been extensively studied for this gene. lacz has nucleotides and its transcription is controlled by the lac operon. in this simulation, transcription is not repressed. thus, provided that the promoter is available for transcription, the expected time for a transcription event to start is approximately s, given the value of the rate constant of reaction in table and that there are rnap molecules available in the system. the promoter open complex formation step, with a mean duration of s and a standard deviation of s is the major limiting factor of transcription events in these conditions. chemical reactions, rate constants, and time delays used to model transcription initiation, elongation, and termination. parameter values were obtained from measurements in e. coli, mainly for lacz. references are reported in the column ref. dynamics of production of proteins comparing the dynamics of the model of translation with measurements recently, the real-time expression of a lac promoter was directly monitored in e. coli with single-protein resolution. the proteins were found to be produced in bursts, with the distribution of intervals between bursts fitting an exponential distribution, while the number of proteins per burst followed a geometric distribution. these distributions were measured for a gene that was kept strongly repressed and for which the ribosome binding site was engineered so that translation was also very weak. under these conditions, our model reproduces these dynamics. nevertheless, we note that it is possible to match these measurements with a simpler model than the one proposed here, where transcription and translation are modeled as single step events. we next compare the kinetics of translation in our model with measurements of the translation elongation speed in three engineered e. coli strains designed to enhance queue formation and traffic in translation. each strain contains a different mutant of lacz. the pmas strain corresponds to the wild-type lacz. the other two sequences differ in that a region of slow-to-translate codons was inserted. the speed of protein chain elongation was measured by subjecting the cells to a pulse of radioactive methionines, and then measuring the level of radioactivity in cells of each population, every s after the pulse. each strand contained methionines, spread out unevenly on the dna sequence, causing the incorporation curve to be non-linear. given that they differ in the nucleotide sequence, it was hypothesized that the translation elongation speed of the three strands would differ, as the speed of incorporation of an amino acid depends on which synonymous codon is coding for it. the cells where translation is faster will thus be expected to have higher levels of radioactivity in the translated proteins, as more labeled amino acids have been incorporated in a fixed time interval. if the translation speeds of the three strands were identical, they would exhibit identical levels of radioactivity at the same point in time. to model this, we simulate the transcription and translation processes of the three sequences. we model the incorporation of radioactive methionines at the same locations as in these sequences. the three model strands differ only in sequence, as in the measurements. during the simulations, we measure the number of incorporated radioactive methionines at the same points in time as in the experiment. results of our simulations and of the measurements are shown in figure showing good agreement between model and measurements. propagation of fluctuations in rna levels to protein levels we simulate the model for varying effective rates of transcription initiation. this rate is determined by the basal rate of transcription initiation, which sets the binding affinity of the rnap to the transcription start site, and by the strength of repression of transcription. thus, to vary keff, we vary the number of repressor molecules present in the system. three sets of simulations are performed, differing in rate of translation initiation. this rate is one of the kinetic parameters of the model, thus can be changed directly, and not by indirect means as keff. in e. coli genes, this rate is believed to be determined by the rbs sequence. mrna and protein degradation rates are set so that the mrna and protein mean levels are identical for all cases, allowing us to study how the level of noise in mrna and protein levels changes. for each set of values of keff and ktr we perform independent simulations. depending on these rates, the mean time to reach steady state differs. each case is simulated for long enough to reach steady state and for an additional s after that. the time series of the simulations for each set of parameter values is concatenated into one time series, from which the noise is quantified by the square of the coefficient of variation, cv. this number of long simulations is necessary to properly sample the system due to the stochasticity of the underlying processes. in figure we first show the cv of mrna time series for varying keff. noise decreases as keff increases due to the promoter open complex formation step. without this event, the distribution of time intervals between transcription initiation events would be exponential, and the cv would not vary. however, with this step, if the expected time for an rnap to bind to the free promoter is faster than the duration of the promoter open complex formation, then the distribution of time intervals becomes gaussian-like. no measurements have yet been made to study experimentally the relation between the noise in mrna levels and the corresponding protein levels. nevertheless, it is possible to create a robust estimate, provided reasonable assumptions on the nature of the underlying processes. our model allows for a direct assessment, and it additionally includes realistic events such as rnap and ribosome traffic, in transcription and translation elongation, which are not included in the aforementioned estimations. figure shows the noise in protein levels, for varying keff and three values of ktr. the data was obtained from the same simulations used to generate the results in figure in general, we find that increasing keff decreases the noise in protein levels due to the decrease of noise in mrna levels. increasing ktr increases the noise in protein levels, due to the increased size of the bursts in the protein level. this finding has not yet been experimentally validated by direct means. an interesting observation from figures and is that, for keff < × 10- s- as keff is increased, the noise in protein levels decreases significantly, while the noise in rna levels does not noticeably change. this is due to the decrease in mean protein burst size, i.e., the mean number of proteins produced from each rna molecule, as both keff and the degradation rate of rna molecules are varied. from these results, we conclude that the degree of coupling between transcription and translation is likely to be a key determining factor of the noise in protein levels. this can be verified by computing the normalized maximum correlation between time-series of protein and mrna levels for each set of parameter values. comparing figures and we see that higher correlation values are obtained for the regime of higher noise in the protein levels. this implies that the principal source of this noise is the fluctuations in rna levels. the correlation value is largely determined by the rates of mrna and protein degradation and production. for example, both increasing the mrna degradation rate and/or decreasing the protein degradation rate increases the time averaging constant of the mrna fluctuations, and thus decreases the correlation between mrna and protein levels. in general, if the mean mrna and protein levels and kept unchanged by tuning their degradation rates accordingly, the correlation between rna and protein time series can be increased by lowering the mrna production rate and/or increasing the protein production rate. effects of transcriptional pauses on the fluctuations in protein levels recent work reported that long transcriptional pauses enhance the noise in mrna levels. we next investigate to what extent the fluctuations in rna levels caused by long transcriptional pauses propagate to protein levels. long sequence-dependent pauses in transcription elongation may cause the ribosome to stall in the mrna chain. this will likely cause subsequent ribosomes to accumulate in the preceding sequence. when the rnap is spontaneously released from the pause, translation of the stalled ribosomes likely resumes but the distribution of intervals between them will differ significantly from what it would have been without the pause event. consequently, the protein production is likely to become burstier, especially if the long pause site is located near the end of the sequence. an increase in burstiness ought to increase the noise in protein levels. to verify this, we perform two simulations. we introduce a long-pause sequence with mean pause durations of s in one case, and s in the other. in both cases, we set the probability that an rnap will pause at that site to 70%. measuring the protein noise levels, we find that the cv is ~5% higher for the s pause site and ~10% higher for the s pause site, in comparison to the same sequence without any sequence specific long-pause site. these relative differences can be biologically relevant in that such a change may, in some cases, cause the degree of phenotypic diversity of a monoclonal cell population to change. the effects of several pause sites on the same strain are cumulative, namely, the higher the number of pause sites, the higher the noise in rna levels. combined with the present results, this leads us to the conclusion that the sequence-dependent transcriptional pausing mechanism likely exists to allow a wide variation of both rna and protein noise levels.  we proposed a new delayed stochastic model of prokaryotic transcription and translation at the single nucleotide and codon level, where the processes of transcription and translation are dynamically coupled in that translation can initiate immediately upon the formation of the ribosome binding site region of the nascent mrna. simulations of the model's dynamics show that, within realistic parameter values, the protein noise levels are determined, to a great extent, by the fluctuations in the rna levels, rather than from sources in translation, in agreement with indirect measurements, as translation elongation was found to be less stochastic than transcription elongation. specifically, the distributions of intervals between translation initiation and translation completion events only differ significantly if the sequence possesses long sequence-dependent pauses or clusters of slow-to-translate codons. the sequence dependence of several mechanisms that can act as generators of strong fluctuations in rna levels, the propagation of these fluctuations to protein levels, and the ability of fluctuations in protein levels to affect cellular phenotype, suggest that these mechanisms may be evolvable. as a previous study has suggested, the translation initiation rate was found to be key in determining the degree of coupling between the fluctuations in rna and protein levels, if one assumes that the degradation rate of the proteins is changed accordingly to maintain their mean level unchanged. varying this sequence-dependent, and thus, evolvable parameter within realistic ranges gave a widely varying degree of coupling between the fluctuations in rna and protein levels. it is therefore not necessarily true that noisy production of rna molecules results in noisy protein levels. interestingly, while decreasing the coupling between transcription and translation by decreasing the rate of translation initiation causes the protein levels to become less noisy, it also takes longer for a change in rna levels to be followed by the protein levels. this suggests that to be able to change rapidly in response to, e.g., environmental changes, the levels of a protein will be necessarily noisier. confirming previous studies, we found that the distributions of time intervals between transcription initiation and completion events differ significantly and that the faster the rate of transcription initiation events, the more they differ. this implies that in the regime of fast transcription, both the transcription and translation elongation processes need to be modeled explicitly and coupled, if one is to match the mean and fluctuations in the protein levels at the molecular level. this is of relevance, since bursts in protein levels may trigger many processes, such as phenotypic differentiation. a final justification for using the model proposed here is the complexity of the process of gene expression in e. coli, and the fact that many events therein may or may not affect the temporal rna and protein levels significantly, depending on their specific sequence-dependent features. such effects, due to the complexity of the system, are not easily predictable without performing explicit numerical simulations. the model proposed here includes several features not included in previous models such as a gradual degradation event that can be triggered while the rna is still being transcribed. as its parameter values were extracted from measurements, it should be useful in the study of several aspects of the dynamics of gene expression in prokaryotes that cannot yet be measured directly and to explore the state space of gene expression dynamics by varying any of the physical variables within realistic ranges. however, the present model does not yet account for known effects of ribosomes on the dynamics of transcription elongation. these might need to be included in future developments of the proposed model as recent results suggest that the rate of translation elongation can affect the rate of transcription elongation, due to possible interactions between the ribosome that first binds to the mrna and the rnap transcribing it. possible effects may include facilitating the release of paused rnap's, which could affect the degree of the contribution of pauses to the noise in rna and thus protein levels. we do not exclude the possibility that the contrary may occur in specific cases, that is, that the paused state of the rnap may cause pauses in the ribosome translational dynamics, which would amplify the effect of transcriptional pauses on the fluctuations of protein levels. whether the pause is ubiquitous or due to loop formations in the nascent rna may affect the results of the interaction as well. provided experimental evidence on the nature and consequences of these interactions, once included in the model, we may be able to test, among other things, whether long transcriptional pauses located in an attenuator system provide an additional layer of control over premature transcription terminations, and thus over rna and protein noise levels. 
to study how events in transcription elongation affect translation elongation and fluctuations in protein levels, we propose a delayed stochastic model of prokaryotic transcription and translation at the nucleotide and codon level that includes the promoter open complex formation and alternative pathways to elongation, namely pausing, arrests, editing, pyrophosphorolysis, rna polymerase traffic, and premature termination. for parameter values within realistic intervals, transcription and translation are found to be tightly coupled in escherichia coli, as the noise in protein levels is mostly determined by the underlying noise in rna levels. finally, modeling sequence-specific transcriptional pauses, we find that these affect protein noise levels. stepwise translation can start after the ribosome binding site is formed and accounts for variable codon translation rates, ribosome traffic, back-translocation, drop-off, and trans-translation. first, we show that the model accurately matches measurements of sequence-dependent translation elongation dynamics. next, we characterize the degree of coupling between fluctuations in rna and protein levels, and its dependence on the rates of transcription and translation initiation. pauses, are found to cause tangible effects in the degree of fluctuations in protein levels. sequence-dependent events in transcription elongation, e.g.
 in prokaryotes, transcription and translation are dynamically coupled, as the latter starts before the former is complete. also, from one transcript, several translation events occur in parallel. to study how events in transcription elongation affect translation elongation and fluctuations in protein levels, we propose a delayed stochastic model of prokaryotic transcription and translation at the nucleotide and codon level that includes the promoter open complex formation and alternative pathways to elongation, namely pausing, arrests, editing, pyrophosphorolysis, rna polymerase traffic, and premature termination. stepwise translation can start after the ribosome binding site is formed and accounts for variable codon translation rates, ribosome traffic, back-translocation, drop-off, and trans-translation.  first, we show that the model accurately matches measurements of sequence-dependent translation elongation dynamics. next, we characterize the degree of coupling between fluctuations in rna and protein levels, and its dependence on the rates of transcription and translation initiation. finally, modeling sequence-specific transcriptional pauses, we find that these affect protein noise levels.  for parameter values within realistic intervals, transcription and translation are found to be tightly coupled in escherichia coli, as the noise in protein levels is mostly determined by the underlying noise in rna levels. sequence-dependent events in transcription elongation, e.g. pauses, are found to cause tangible effects in the degree of fluctuations in protein levels. 