Bull Math Biol (2014) 76:314–334 DOI 10.1007/s11538-013-9920-8 O R I G I N A L A RT I C L E
Optimal Performance of the Tryptophan Operon of E. coli: A stochastic, Dynamical, Mathematical-Modeling Approach Emanuel Salazar-Cavazos · Moisés Santillán
Received: 24 May 2013 / Accepted: 7 November 2013 / Published online: 4 December 2013 © Society for Mathematical Biology 2013
Abstract In this work, we develop a detailed, stochastic, dynamical model for the tryptophan operon of E. coli, and estimate all of the model parameters from reported experimental data. We further employ the model to study the system performance, considering the amount of biochemical noise in the trp level, the system rise time after a nutritional shift, and the amount of repressor molecules necessary to maintain an adequate level of repression, as indicators of the system performance regime. We demonstrate that the level of cooperativity between repressor molecules bound to the first two operators in the trp promoter affects all of the above enlisted performance characteristics. Moreover, the cooperativity level found in the wild-type bacterial strain optimizes a cost-benefit function involving low biochemical noise in the tryptophan level, short rise time after a nutritional shift, and low number of regulatory molecules. Keywords Gene network regulation · Stochastic gene expression · Dynamic optimization · Thermodynamic optimization 1 Introduction Tryptophan is an essential amino acid, and thus it cannot be produced by humans and other animals. However, it can be synthesized by plants and microorganisms; particularly by E. coli. Tryptophan is also the heaviest and the most biochemicallyexpensive amino acid to synthesize. As a matter of fact, it is one of the less abundant
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E. Salazar-Cavazos · M. Santillán ( ) Centro de Investigación y Estudios Avanzados del IPN, Unidad Monterrey, Apodaca NL, México e-mail:
[email protected] E. Salazar-Cavazos University of New Mexico Health Sciences Center, Department of Pathology, Albuquerque, NM, USA
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amino acids in all living species (Lehninger et al. 2008). These facts explain why the tryptophan-biosynthesis metabolic pathway in the organisms capable of synthesizing this amino acid is, invariably, tightly regulated. We are interested in investigating whether the parameters of the tryptophan biosynthesis regulatory mechanisms have been tuned by evolution to optimize some objective function. Previous works (Angulo-Brown et al. 1995, 2001; Santillán and Angulo-Brown 1997; Santillán et al. 1997; Klipp et al. 2002; Wall et al. 2003; Dekel and Alon 2005; van Hoek and Hogeweg 2007; Páez-Hernández et al. 2006; Páez-Hernández and Santillán 2008; Mackey and Santillán 2008; Oyarzún et al. 2009; Díaz-Hernández et al. 2010; Poelwijk et al. 2011; Quan et al. 2012; Chubukov et al. 2012), in which the performance regimes of several biological and nonbiological systems are analyzed from an optimization perspective, suggest that an optimal performance regime generally involves a trade-off of dynamical and energetic characteristics. Some of these works, for instance, focus on thermodynamic properties like power and efficiency, and show that, usually, one increases while the other decreases in many types of biological systems (Angulo-Brown et al. 1995; Santillán and Angulo-Brown 1997; Santillán et al. 1997). Other works demonstrate that often times the dynamic stability and the thermodynamic power output of biological and non biological systems are controlled by the same parameters, and that they cannot be optimized simultaneously but a tradeoff has to be achieved (Angulo-Brown et al. 2001; Páez-Hernández et al. 2006; Páez-Hernández and Santillán 2008; Mackey and Santillán 2008; Díaz-Hernández et al. 2010). Finally, Oyarzún et al. (2009), Poelwijk et al. (2011), Quan et al. (2012), Chubukov et al. (2012) are advocated to studying the objective functions optimized by the performance regimes of various gene regulatory networks and metabolic pathways. Inspired on these antecedents, we hypothesize that, in the specific case of tryptophan, the corresponding biosynthesis regulatorypathway (also known as the tryptophan operon) has evolved in such a way that: low noise is observed in the intracellular tryptophan levels, the system dynamics adapt with short rise times after environmental changes, and the regulatory mechanisms demand a low level of biochemical energy consumption. The present paper is advocated to testing this hypothesis from a mathematical modeling viewpoint. The paper is organized as follows. In Sect. 2, we introduce all of the relevant biological concepts and definitions concerning the tryptophan operon and its regulatory pathway; Sect. 3 is advocated to the development of the mathematical model, the estimation of the model parameters, and the introduction of the employed numerical methods; all the obtained results are presented and discussed in Sect. 4; and finally, some concluding remarks are given in Sect. 5.
2 Concepts and Definitions The amino acid tryptophan can be synthesized by plants and bacteria, but not by humans and other mammals. In E. coli, the polypeptides constituting the enzymes in the tryptophan biosynthesis pathway are encoded by the so-called tryptophan operon genes: trpE, trpD, trpC, trpB, and trpA. These genes are transcribed from trpE to trpA, and transcription starts at promoter trpP—located upstream from gene trpE.
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Three different negative-feedback mechanisms regulate the trp operon: repression, transcription attenuation, and enzyme inhibition. We briefly review these mechanisms below, based on Yanofsky and Crawford (1987), Yanofsky (2000), Xie et al. (2003), Grillo et al. (1999), Jeeves et al. (1999), Brown et al. (1999). Repression is mediated by three operators (O1 , O2 , and O3 ) overlapping with the operon promoter, trpP. When active repressors bind one or more operators, they prevent RNA polymerases from binding to the promoter, and thus avoid transcription initiation. The trp repressor is a dimeric protein that becomes active when its two tryptophan binding sites are occupied. Thus, the higher the intracellular tryptophan level, the larger the amount of active repressors and the higher the promoter repression level. Finally, when operators O1 and O2 are occupied, the corresponding repressors interact cooperatively increasing the complex stability. Transcription attenuation consists of a premature termination of mRNA transcription. The DNA region between the tryptophan promoter and gene trpE, also called the leader region, is responsible for attenuation control. The leader-region transcript consists of four segments, termed Segments 1, 2, 3, and 4. Once the first two segments are transcribed, they form a hairpin that stops transcription. When a ribosome starts translating the nascent mRNA, it disrupts Hairpin 1:2 and transcription is restarted together with translation. Segment 1 has two tryptophan codons in tandem. Thus, if tryptophan is scarce, and so are loaded tRNATrp ’s, the ribosome remains in the first segment while transcription continues. This facilitates the development of Hairpin 2:3 (the antiterminator), and transcription proceeds into the structural genes. Contrarily, if tryptophan is abundant, the ribosome rapidly finishes translation of Segments 1 and 2, and eases the formation Hairpin 3:4, which is recognized by RNA polymerase as a termination signal and so transcription is aborted. Enzyme inhibition takes place through anthranilate synthase. This enzyme catalyses the first and slowest reaction in the tryptophan biosynthesis catalytic pathway. Anthranilate synthase is a hetero-tetramer made up of two TrpE and two TrpD polypeptides, and it becomes inhibited when the TrpE subunits are bound by tryptophan molecules. Hence, an excess of intracellular tryptophan inactivates this enzyme and avoids further tryptophan synthesis.
3 Methods 3.1 Model Development We developed a model to study the stochastic dynamics of the tryptophan operon of E. coli. The variables of interest are the state of the promoter, as well as the mRNA, anthranilate synthase enzyme, and tryptophan molecular counts. All the other processes taking place in the system like: repressor activation, enzyme inhibition, transcriptional attenuation, etc. are assumed to obey equilibrium deterministic kinetics because they take place at much faster rates, and so there is a natural separation of time scales (Hernández-Valdez et al. 2010; Salazar-Cavazos and Santillán 2012). These fast processes are implicitly taken into consideration within the slow-reaction effective propensities (Zeron and Santillán 2010). The model also accounts for time
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Fig. 1 Schematic representation of the processes involved in the tryptophan operon regulatory pathway. Dij k represents the promoter binding state, i, j, k = 0, 1 if the first, second and third operators are respectively empty or bound by an active repressor. Solid arrows denote the processes in this pathway: promoter switching between binding states, mRNA synthesis and degradation, enzyme synthesis and degradation, and tryptophan (Tryp) production and consumption. Dashed arrows and hammer-heads represent either positive or negative influence of system chemical species onto the processes they point to. Although the effect of tryptophan on promoter switching is represented with a single line for the sake of clarity, it must be pointed out that all the promoter state-switching reactions are affected
delays associated to transcription and translation. A detailed description of the model development process is given below. A schematic representation of the processes accounted for by the model is given in Fig. 1. As previously discussed, repressor molecules are activated when they are bound by a couple of tryptophan molecules. The kinetics of repressor activation were analyzed in Santillán and Zeron (2004), where the number of active repressors is demonstrated to be given by 2 T , (1) R2T = RTot T + KT in which RTot stands for the total number of repressor molecules, KT for the dissociation constant between tryptophan and one binding site of a repressor, and T for the tryptophan molecule count. There are three different repressor binding sites (operators) overlapping the trp promoter. Hence, the promoter can be in eight different states, with each operator being either free or bound by a repressor molecule. Furthermore, when two repressor molecules are bound to the first and second operators, they do it cooperatively (Grillo et al. 1999). To model promoter dynamics, we assume that the binding processes are independent. That is, although all binding propensities are proportional to the number of active promoters, the binding propensity to a specific operator does not depend on the state (free or bound) of the two other operators. Hence, the binding propensity to operator Oi can be written as ki+ R2T ,
(2)
where i = 1, 2, 3, ki+ are parameters measuring the operator affinities to active promoters, and R2T the active promoter count.
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Let ki− denote the detachment propensity from operator Oi when all other operators are empty. Since operator O3 does not interact cooperatively with the other two operators, its detaching propensity is always k3− , regardless of the states of O1 and O2 . For the same reason, the detachment propensities from operators O1 and O2 do not depend on the state of operator O3 . Finally, to account for the cooperativity between O1 and O2 , we suppose that whenever both of them are bound by active repressors, the detachment propensity from operator O1 is k1− /kc , while the detachment propensity from O2 is k2− /kc . The value of parameter kc > 1 reflects the level of cooperativity between these two operators. One or the two tryptophan molecules bound to an active repressor can detach from it while it is bound to any of the operators, and this decreases the stability of the repressor-operator complex (Grillo et al. 1999). To take this into account we assumed that the value of parameters ki− depends on the intracellular tryptophan level as follows: − − − ki− = ki,R P (Oi : R) + ki,R P (Oi : RT ) + ki,R P (Oi : R2T ), T 2T
(3)
− − − in which ki,R , ki,R , and ki,R , respectively, represent the propensities of a repressor T 2T dissociating from operator i when the repressor has none, one, or two tryptophan molecules bound to it. Furthermore, P (Oi : R), P (Oi : RT ), and P (Oi : R2T ) denote the probabilities of having the repressor bound by none, one, or two tryptophans when it is bound to operator Oi . The reactions through which tryptophan molecules bind to and detach from a repressor-operator complex are KB
KB
Oi : R2T Oi : RT + T Oi : R + 2T ,
(4)
where KB is the corresponding dissociation constant. By assuming chemical equilibrium and working out with the resulting chemical kinetics equations we get 2 KB P (Oi : R) = , KB + T 2(KB )(T ) , (KB + T )2 2 T . P (Oi : R2T ) = KB + T P (Oi : RT ) =
(5)
Due to transcriptional attenuation, only a fraction of the polymerase molecules that initiate transcription reach the end of the trp genes and produce functional mRNA molecules, which in turn are translated to produce the proteins coded by the trp genes. Santillán and Zeron (2004) found that the probability that transcription is not prematurely terminated due to transcriptional attenuation is PA (T ) =
1 + 2α KGT+T (1 + α KGT+T )2
with α and KG being parameters to be estimated.
(6)
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If we denote the promoter state as Dj,k,l , with j, k, l = 0, 1 if the first, second, and third operators are respectively empty or bound by an active repressor, the only state from which transcription can start is D0,0,0 . On the other hand, several modeling studies suggest that transcription can be modeled as a single chemical step; see, for instance, Shahrezaei and Swain (2008) and references therein. Therefore, in the present work, we model transcription through the following reaction: kM
D0,0,0 −→ D0,0,0 + M,
(7)
representing the propensity of mRNA synthesis with M denoting mRNA and kM from a non-repressed promoter. From the considerations in the previous paragraph, is modulated by transcriptional attenuation. Hence, the value of kM = kM PA (T ), kM
(8)
with kM the propensity of transcription initiation at a non-repressed promoter. Although transcription is modeled as a single step process, it is not regarded as instantaneous. We assume that, after transcription initiation, it takes a time τM to have a mRNA chain long enough to allow translation to occur without being halted by the progress of RNA polymerase. In the present model, mRNA degradation is assumed to take place according to the following reaction: γM
M −→ ∅,
(9)
with the effective degradation propensity, γM , accounting for both mRNA degradation and dilution due to bacterial cell growth. To account for dilution due to cell growth, in the present work we consider a constant volume cell and assume that all molecular species are lost at a rate μ that equates the bacterial growth rate. Of all the proteins coded for by the trp genes, only TrpE is important from a regulatory perspective. The first reaction in the biochemical pathway leading to tryptophan biosynthesis is catalyzed by anthranilate synthase, which is a hetero-tetramer made up of two TrpE and two TrpD subunits. This reaction is also the slowest one along the tryptophan biosynthesis pathway, and so it determines the speed of the whole process. Finally, anthranilate synthase is subject to feedback enzyme inhibition when two tryptophan molecules bind the TrpE subunits. Assume that the polymerization process is fast enough so that the count of anthranilate synthase molecules is approximately one half that of TrpE molecules. Let further E denote an anthranilate synthase molecule. From the previous discussion, anthranilate synthase production can be modeled as kE /2
M −→ M + E,
(10)
where kE is the translation initiation propensity at the trpE ribosome binding site. The assumption that translation can be modeled as a single step process is implicit in the above reaction. Finally, we suppose that translation is not instantaneous, but that a time τE elapses between translation initiation and the delivery of a functional protein.
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Enzyme degradation is modeled through the following reaction: γE
E −→ ∅.
(11)
In principle, the effective protein degradation propensity, γE , takes into account both protein degradation and dilution due to bacterial cell growth. However, following Bliss et al. (1982) and Sinha (1988), we assume a negligible degradation rate for this enzyme because, according to their estimates, it is much smaller than the growth rate. We further consider that tryptophan is produced and consumed through the following reactions: kT
EA −→ EA + T , γT
T −→ ∅,
(12) (13)
with T denoting a tryptophan molecule, and EA an active enzyme. It must be emphasized that in this case the effective tryptophan consumption propensity, γT , is not a simple constant because tryptophan consumption is a catalytic process. As described above, anthranilate synthase enzymes are inhibited when a couple of tryptophan molecules bind the enzyme TrpE subunits, and so the amount of active enzymes depends on the tryptophan concentration. Following Caligiuri and Bauerle (1991), the amount of active anthranilate synthase in the active state can be approximated by EA (T ) = E
KIn , KIn + T n
(14)
with KI the corresponding half-saturation constant, and n a Hill coefficient. Regarding tryptophan consumption, Hernández-Valdez et al. (2010) demonstrated that its consumption rate can be modeled as γ =γ
T , T + Kρ
(15)
with γ the maximum propensity of tryptophan consumption and Kρ the trp consumption half- saturation constant. From the discussion in the previous paragraphs, the tryptophan operon regulatory pathway can be visualized as the set of chemical reactions tabulated in Table 1, together with their corresponding propensities. Observe that the reaction propensities depend on the system state. As a matter of fact, the system feedback regulatory loops are taken into account into the reaction propensities. Repression is accounted for by Eq. (2), transcriptional attenuation is considered by Eq. (8), and Eq. (14) takes into account enzyme inhibition. 3.2 Parameter Estimation We paid special attention to the estimation of all the model parameters from reported experimental data. The parameter values we employ in the present work and the detailed procedure to estimate them are given in the Appendix.
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Table 1 List of all the reactions accounted for by the present model, together with their corresponding propensities. Note that the same symbols are employed to denote chemical species (in the chemicalreaction columns), and the corresponding amounts (in the propensity columns). The first 12 rows correspond to the promoter switching among the available binding states, row 13 accounts for the reactions through which mRNA is produced and degraded, the penultimate row accounts for enzyme synthesis and degradation, and the final row comprises the reactions through which tryptophan is produced and consumed Reaction
Propensity
Reaction
Propensity
D111 D110
k3− (T )D111
D110 D111
k3+ D110 R2T (T )
D111 D101 D111 D011 D110 D100 D110 D010 D101 D100 D101 D001 D011 D010 D011 D001 D100 D000 D010 D000 D001 D000
k2− (T )D111 /kc k1− (T )D111 /kc k2− (T )D110 /kc k1− (T )D110 /kc k3− (T )D101 k1− (T )D101 k3− (T )D011 k2− (T )D011 k1− (T )D100 k2− (T )D010 k3− (T )D001
D101 D111 D011 D111 D100 D110 D010 D110 D100 D101 D001 D101 D010 D011 D001 D011 D000 D100 D000 D010
k2+ D101 R2T (T ) k1+ D011 R2T (T ) k2+ D100 R2T (T ) k1+ D010 R2T (T ) k3+ D100 R2T (T ) k1+ D001 R2T (T ) k3+ D010 R2T (T ) k2+ D001 R2T (T ) k1+ D000 R2T (T ) k2+ D000 R2T (T )
D000 D001
k3+ D000 R2T (T ) γM M
D000 D000 + M
kM D000 PA (T )
M ∅
M M +E
kE M/2
E∅
EA EA + T
kT EA (T )
T ∅
γE E
γ (T )
3.3 Numerical Methods The time evolution of the reaction network that models the tryptophan operon regulatory pathway was simulated by means of a variation of the celebrated Gillespie algorithm (Gillespie 1977), which was developed by Cai (2007) and accounts for time delayed reactions. This algorithm was implemented in Python.
4 Results We started by simulating the so-called de-repression experiments carried out by Yanofsky and Horn (1994). In these experiments, an E. coli culture is initially grown in a medium with abundant tryptophan. In consequence, the trp genes are completely repressed and the existing anthranilate synthase enzymes are all inhibited. Thereafter, the bacterial cells are washed, shifted to a medium with no tryptophan, and allowed to grow there for a long time. Shortly after the medium shift, all the tryptophan negative-feedback regulatory mechanisms are released and the operon genes start being expressed, until they eventually reach a stationary expression level. Yanofsky and Horn (1994) periodically measured the activity of enzyme anthranilate synthase to investigate the dynamics of the trp operon after a nutritional shift. To mimic the de-repression experiments of Yanofsky and Horn, we carried out stochastic simulations with the model introduced in the Methods section, choosing as
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Fig. 2 Results of a stochastic simulation with a wild type strain for the mRNA, enzyme, and tryptophan counts, and the corresponding histograms
initial state that in which the promoter is fully repressed (all the operators are bound by active repressors), there are no messengers, the number of enzyme molecules is 50, and within each cell there exist 80,000 tryptophan molecules. This initial condition is meant to reproduce the state of a bacterium that has grown in the presence of abundant tryptophan for a long time. In Fig. 2, we present the results of one of such stochastic simulations for the wild type strain, together with the histograms for the mRNA, enzyme, and tryptophan molecular counts. Even though we only show the first 100 min of the stochastic sim-
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Fig. 3 Plots of enzyme count vs. time resulting from averaging 1000 model simulations carried out as described in the main text (solid lines), compared with two different sets of experimental results (solid circles and triangles) by Yanofsky and Horn (1994) for the wild type strain (panel A), and with one set of experimental results for the inhibition-less strain (panel B)
ulation, the histograms were computed from time series 5,000 points long (1,000 min). Thereafter, we took the results of 1000 of the previously described simulations, averaged them, and compared with the experimental results of Yanofsky and Horn (1994). In Fig. 3A, we show the results for a wild-type bacterial strain, while in Fig. 3B, we show the results for an inhibition-less strain. This last strain contains a mutation in gene trpE such that, although functional, the enzyme anthranilate synthase cannot be feedback inhibited by tryptophan. We simulated the inhibition-less strain by setting the probability that an enzyme is not inhibited equal to one. Observe that, in both cases, there is a good agreement between the model simulations and the experimental data. This makes us confident that our model captures the essential dynamic features of the tryptophan operon with enough details so as to make further predictions. Our next step consisted in investigating the effect of the different existing regulatory mechanisms on the system biochemical noise. To this end, we carried out simulations with the wild-type and the following hypothetical mutant strains: • Repression-less strain, constructed by setting the total number of repressor molecules equal to zero. • Cooperativity-less strain, constructed by setting the cooperativity constant kc equal to one. • Attenuation-less strain, constructed by setting the probability that a transcriptional event is not prematurely terminated PA = 1. • Inhibition-less strain, constructed by setting up the number of active enzymes equal to the total number of enzymes EA = E. To estimate the level of biochemical noise we measured the coefficient of variation (CV) from the mRNA, the enzyme, and the tryptophan time series resulting from the previously delineated stochastic simulations. We took care of measuring the coefficients of variation after the corresponding average values have reached their stationary values (see Fig. 3), and employed time series 10,000 data points (2,000 min) long.
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We performed these measurements in the wild-type and the mutant bacterial strains described above. It is known that the coefficient of variation strongly depends on the average molecule count. As a matter of fact, in an ideal birth-death process in which molecules are produced via a Poisson process and the degradation propensity is proportional to the molecule count, the coefficient of variation is given by (Shahrezaei and Swain 2008; Raj and van Oudenaarden 2009) CV Ideal = 1/ N , with N the stationary value of the mean number of molecules. The previously described mutant bacterial strains are expected to have different average molecular counts, and this would suffice to explain observed differences in the coefficients of variation. However, it is also possible that the removal of a given regulatory mechanism affects biochemical noise independently of changes in the average number of molecules. We are interested in this last phenomenon and so, in order to quantify it, we define a normalized coefficient of variation as follows: cv = CV/CV ideal . From its definition, cv measures how much larger the current amount of biochemical noise is as compared with that of an ideal birth-death process with the same average number of molecules. The obtained cv values for all the previously described mutant strains and for all chemical species are shown in Figs. 4A–C. Observe that, contrarily to what one would naively expect, the amount of noise at the mRNA and enzyme levels are not correlated (see Figs. 4B and 4C). In fact, all bacterial strains show similar amounts of noise in the enzyme count, except for the repression-less strain which is appreciably noisier. We believe that these behaviors can be explained as follows. All bacterial strains with repression have similar noise levels in the enzyme count despite having different noise levels in the messenger numbers because the enzymes have much larger half lives than mRNA’s. In other words, at the enzyme half-life time scale, the much faster mRNA fluctuations are averaged out. This is in agreement with the experimental and theoretical results of Taniguchi et al. (2010), who measured the expression of several E. coli genes at the single molecule level, as well as with the common notion in engineering that systems with long characteristic times work as low-pass filters (Cartwright et al. 2012). On the other hand, the fact that removing repression greatly increases enzyme biochemical noise is in complete agreement with the common knowledge that negative feedback loops serve, among other things, to reduce noise (Dublanche et al. 2006). Transcriptional attenuation is also a negative feedback regulatory loop. However, it has no noticeable noise-reduction effects, possibly because it is fine-tuned to only respond under conditions of extreme tryptophan starvation (Yanofsky et al. 1984; Yanofsky 2000). The tryptophan-count noise levels are shown in Fig. 4C, and zoomed-in in Fig. 4D. A comparison of Figs. 4B and 4D reveals that, except for the inhibition-less strain, noise at the tryptophan level has a similar tendency to noise at the enzyme level: larger
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Fig. 4 Plots of the mRNA (A), enzyme (B), and tryptophan (C) normalized noise levels, computed from stochastic simulations of the following bacterial strains: cooperativity-less (Coop-less), repression-less (Rep-less), wild-type (WT), inhibition-less (Inh-less), and attenuation-less (Att-less). The plot in panel D is the same as in panel C but at a different scale
enzyme noise means larger tryptophan noise. On the other hand, tryptophan noise is in all cases more than twice as large as enzyme noise. These observations can be explained by the facts that tryptophan dynamics is much faster than enzyme dynamics (explaining why noise propagates from the enzyme to the tryptophan levels), and that every enzyme produces several tryptophan molecules (explaining noise amplification
326 Table 2 Characteristics of the hypothetical mutant strains constructed to investigate the dynamical effects of repressor cooperativity. WT stands for wild type
E. Salazar-Cavazos, M. Santillán Strain
Cooperativity Constant
Total Repressor Count
hms1
792
hms2
185
200
WT
40
400
hms3
7
800
hms4
1
1320
100
when going from enzymes to tryptophan). In the case of the inhibition-less strain, the tryptophan noise is about two orders of magnitude larger than enzyme noise. This is also in agreement with the fact that negative feedback loops help to reduce noise (Dublanche et al. 2006). Recall that enzyme inhibition is the regulatory feedback loop immediately before tryptophan synthesis. We have seen that repressor cooperativity increases noise in the mRNA count. However, this noise increment is not transferred to the enzyme and tryptophan levels, possibly due to the large half life of enzymes as compared to that of mRNA. On the other hand, the repression and enzyme inhibition regulatory mechanisms play important roles in curbing noise at the enzyme and tryptophan levels, respectively. Hence, if repressor cooperativity increases mRNA noise but has no effect on the enzyme and trp noise levels, what is the evolutionary advantage this trait confers to E. coli? To investigate this question we constructed various hypothetical mutant strains, with different levels of repression cooperativity. However, in order to make direct comparisons of the noise levels we needed all the strains to possess the same average molecular counts in the stationary state. That is, all of them need to have the same stationary level of repression. To guarantee this, each time the cooperativity level was changed, we modified the total repressor count accordingly. In Table 2, we summarize the characteristics of these novel hypothetical mutant strains. We carried out several 500 min long stochastic simulations for all the hypothetical bacterial strains in Table 2 and computed in each case the amount of noise in the number of tryptophan molecules, and the rise time in a de-repression experiment (defined as the time the average tryptophan level takes to reach 95 % of its stationary value). The results are shown in Fig. 5. One can argue that the objective of the tryptophan operon is to guarantee an adequate level of this amino acid, regardless of variations in the extracellular medium. From that perspective, large tryptophan fluctuations would be deleterious for a bacterium. When the tryptophan count decreases way below the average value, the cell would not have enough amino acid to synthesize all the necessary proteins. On the other hand, if there is over abundance of tryptophan, the cell would be expending important amounts of energy to synthesize unnecessary amino acid molecules; recall that tryptophan is the most expensive amino acid to synthesize. Therefore, a low amount of noise in tryptophan count can be regarded as an advantageous evolutionary trait. Interestingly, Fig. 5A reveals that the less noisy strain is the one denoted as hms3 in Table 2. That is, the strain with twice as many repressor molecules as the wild-type strain. Having a short rise time after a nutritional shift is another evolutionary advantageous trait. According to Bhartiya et al. (2003), reaching an adequate level of intra-
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Fig. 5 Values of tryptophan noise (A), rise time—in minutes—(B), and cost-benefit function—see the main text for its definition—(C), for all the strains tabulated in Table 2
cellular tryptophan shortly after the nutritional shift is beneficial because it allows the cells to rapidly start synthesizing again all the proteins they need to grow, even if the price to pay is a transient overshot of tryptophan production as seen in Fig. 3A. Figure 5B shows that the rise time after a de-repression experiment is a decreasing (increasing) function of the total repressor count (cooperativity level). Thus, from this perspective, the fittest strain would be that with no repressor cooperativity and 1320 repressor molecules. We have seen that bacterial strains with more repressor molecules and less cooperativity have a smaller amount of noise and a shorter rise time than the wild-type strain. Then, why were they not favored by natural selection? In our opinion, the answer is: because a larger number of repressor molecules implies a larger metabolic cost for the cell. From this, we speculate that natural evolution has favored those bacteria that attain a good compromise between low noise in the tryptophan count and short rise time after nutritional shifts, on the one hand, and a low metabolic cost due to production of repressor molecules, on the other hand. To test this hypothesis, we
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defined the following cost-benefit function: Ψ=
R ηhms4 τhms4 + − , η τ Rhms4
(16)
where η, τ , and R, respectively, denote the coefficient of variation of the Trp time series, the rise time after a de-repression experiment, and the total number of repressor molecules of a given bacterial strain, while the sub-index hms4 refers to the hypothetical mutant bacterial strain with no repressor cooperativity. We normalize with respect to this strain based on the assumption that the first bacterium to developed repression (as a regulatory mechanism for the tryptophan operon) lacked cooperativity; and that this feature evolved afterwards because it allowed the cells to reduce the metabolic cost associated to repression. From its definition, maximizing function Ψ implies an optimal trade-off between low noise, short rise time, and low cost associated to the synthesis of repressor molecules. We evaluated function Ψ for the five strains tabulated in Table 2 and the results are plotted in Fig. 5C. Notably, of all the bacterial strains, the one with the highest Ψ value is the wild type strain.
5 Concluding Remarks We developed a detailed mathematical model to study the stochastic dynamics of the tryptophan operon of E. coli. All the model parameters were estimated from reported experimental data. To validate our model, we simulated a couple of dynamic experiments carried out by Yanofsky and Horn (1994), obtaining a good agreement between the experimental results and the model predictions. This made us confident to use the model to investigate different aspects of the tryptophan operon stochastic dynamics. In particular, we were interested in finding out whether this system performance optimizes some kind of cost benefit function. The obtained conclusions are summarized below. Our results suggest that the existing cooperative interaction between operators O1 and O2 increases the mRNA biochemical noise levels. However, repression, a negative feedback regulatory mechanism immediately behind mRNA production, seems to partially control mRNA noise. Moreover, according to our simulations, random fluctuations in the mRNA count does not propagate to the enzyme level, presumably because of the notable difference of time lives existing between enzymes and mRNA molecules. Our results further indicate that, since tryptophan production and consumption are much faster processes than enzyme synthesis and degradation, enzyme noise does propagate to the tryptophan level. However, enzyme inhibition, the negative feedback regulatory mechanism immediately behind tryptophan production, plays a very important and efficient role in reducing tryptophan noise. Finally, we speculate from the analysis of the present model results that the level of cooperativity found in the wild-type strain represents an optimal trade-off between a low cost associated to the production of repressors, and high benefits associated to a short rise time after nutritional shifts and a low amount of biochemical noise in the number of tryptophan molecules.
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Acknowledgements ES-C acknowledges financial support from Consejo Nacional de Ciencia y Tecnología (CONACyT) through scholarship No. 302844. MS thanks McGill University and particularly Prof. Michael C. Mackey for their hospitality during his sabbatical leave, during which part of the research for this paper was performed. Both authors are grateful with the anonymous reviewers whose comments greatly helped to improve this paper.
Appendix: Parameter Estimation In this work, we consider a cell volume of 1 µm3 (Javelle et al. 2005). On the other hand, from experimental measures made by Bennett et al. (2009) on E. coli growing in minimal medium with glucose, the bacterial average doubling time is 77 min. Then μ ≈ 0.013 min−1 . Baker and Yanofsky (1972) calculated that during exponential growth E. coli possess around 1.8 copies of the trp operon. Therefore, the probabilities that a cell has 1 or 2 copies are 0.2 and 0.8, respectively. We developed an algorithm that use these probabilities to randomly choose the initial number of copies of the trp operon, and thus of the promoter. The initial state of the promoter(s) is selected by another algorithm that calculates and uses the probabilities of each of the 8 possible states at the initial levels of tryptophan. From the work of Morse et al. (1968), the number of anthranilate synthase enzymes before de-repression is E0 ≈ 50 molecules. On the other hand, we have from the website E. coli Statistics (http://ccdb. wishartlab.com/CCDB/cgi-bin/STAT_NEW.cgi) that the number of tryptophan molecules is T0 ≈ 80,000 molecules. The dissociation constant of the reaction through which a tryptophan binds one of its binding sites at the repressor, as obtained by Arvidson et al. (1986), is KT ≈ 44,160 molecules. From the experimental results of Gunsalus et al. (1986), in which the number of repressors in E. coli cultured in medium without tryptophan was calculated, we have that RTot ≈ 400 molecules. For the association propensities of the active repressor with the different operators (ki+ ) and the dissociation propensities of the repressor in its three different states − − − , ki,R , and ki,R ), we used the values employed by Tabaka et al. (2008). These (ki,R T 2T values, calculated from the experimental works of Grillo et al. (1999), Hurlburt and
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Yanofsky (1992), Zhang et al. (1994), and Jardetzky and Finucane (2007), are shown below: k1+ ≈ 8.1 molecules−2 min−1 ,
− k1,R ≈ 6.0 molecules−1 min−1 , 2T
− k1,R ≈ 19.2 molecules−1 min−1 , T
− k1,R ≈ 60.0 molecules−1 min−1 ,
k2+ ≈ 0.312 molecules−2 min−1 ,
− k2,R ≈ 0.198 molecules−1 min−1 , 2T
− k2,R ≈ 6.6 molecules−1 min−1 , T
− k2,R ≈ 66.0 molecules−1 min−1 ,
k3+ ≈ 0.3 molecules−2 min−1 ,
− k3,R ≈ 36.0 molecules−1 min−1 , 2T
− k3,R ≈ 72.0 molecules−1 min−1 , T
− k3,R ≈ 810.0 molecules−1 min−1 .
The dissociation constant of the reaction through which a tryptophan molecule binds one of its binding sites in a repressor bound to an operator, KB , was recalculated from the work of Tabaka et al. (2008) using the KT value obtained by Arvidson et al. (1986), instead of the one estimated by Schmitt et al. (1995): KB ≈ 1,980 molecules. The constant accounting for the cooperative interaction between repressors bound to operators O1 and O2 was experimentally estimated by Yang et al. (1996): kc ≈ 40. We computed the propensity for transcription initiation at a non-repressed promoter from the reported transcription initiation rate of the lac operon, and from the comparative strengths of the trp, lacUV5 and lac promoters (Kennell and Riezman 1977; De Boer et al. 1983; Deuschle et al. 1986). The trp promoter resulted to be around 1.43 times stronger than the lac promoter, which has a maximal transcription rate of 18.2 molecules/min. Thus, kM ≈ 26 molecules/min. Yanofsky et al. (1984) found that transcriptional attenuation is relieved only when the intracellular concentration of tryptophan is extremely low, and that the probability that transcriptional attenuation occurs at these conditions is 6 times smaller in comparison to instances in which tryptophan concentration is higher. The values of parameters KG and α that allow Eq. (6) to represent this behavior are KG ≈ 1200 molecules and α ≈ 18.8. To calculate the time between transcriptional initiation and the moment in which translation can start without being momentarily stopped by the RNA polymerase, τM , we need to remember that the enzyme anthranilate synthase is produced from trpE and trpD genes. Knowing that the distance between the site of transcriptional initiation and the start codon of trpD gene is 1724 nucleotides (nt) and that the RNA polymerase in bacteria with a doubling time of 60 min advance at a speed of a speed of
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2700 nt/min, we obtain that it will take 0.64 min for the polymerase to start transcribing gene trpD (Yanofsky et al. 1981; Bremer and Dennis 1996). On the other hand, if we consider that the ribosome advances at a slightly higher rate (2880 nt/min) than the RNA polymerase, and that the length of trpD gene is 1596 nt, we obtain that τM ≈ 0.68 min. Forchhammer et al. (1972) determined that the regions of the mRNA corresponding to the trpE and trpD genes have half-lives of 1 and 1.25 min, respectively. Using the shortest half-life, we have that the mRNA-degradation propensity is γM ≈ 0.69 min−1 . To obtain the translation initiation propensity, kE , we used two experimentally determined values. The first one is the number of ribosomes that translate each mRNA from the trp operon in all its life time, which is approximately 30 (Baker and Yanofsky 1972). The second one is the mRNA mean lifetime, which can be calculated from γM as 1.45 min. Thus, kE ≈ 20.7 molecules/min. Considering that the subunit of the enzyme anthranilate synthase that takes more time to be produced is TrpD, that the length of the trpD gene is 1,596 nt, and the ribosome elongation rate is 2,880 nt/min, we have τE ≈ 0.55 min. The propensity of tryptophan production by active enzyme was obtained from the experimental studies of Ito et al. (1969): kT ≈ 300 molecules/min. The values of the half saturation constant KI and the corresponding Hill coefficient n, where calculated by Caligiuri and Bauerle (1991): KI ≈ 2500 molecules,
and n ≈ 1.2.
To compute the maximum propensity of tryptophan consumption, γ , we took the number of tryptophan molecules incorporated in all the proteins of the cell and we divided it by the time in which the cell, and thus the proteins, are doubled. From the website E. coli Statistics we obtain that each E. coli bacteria has around 2.6 million proteins, each one with an average length of 360 amino acids. With this information, and knowing that the relative abundance of tryptophan with respect to all the amino acids is 1.1 %, there are around 10.3 million tryptophan molecules present in proteins (Neidhardt et al. 1990). If the doubling time is 77 min, then γ ≈ 134,000 molecules/min.
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To take into account that protein production, and consequently tryptophan consumption, decreases only when the number of tryptophan molecules in the cell is scarce we used Kρ ≈ 1,000 molecules. The system described in this work reaches stationary levels for the enzyme and tryptophan of around 1000 and 4100 molecules per bacterial cell, respectively. These values are in agreement with the experimental data of Bliss et al. (1982) and Bennett et al. (2009).
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