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Morgan Price
Morgan PriceNormal.dot:mRNA Degradation and the Fitness of Unicellular OrganismsI)ux
lTT8L<Tz"T8(0lmRNA Degradation and the Fitness of Unicellular OrganismsMorgan Price
November 2002
Abstract
Recent genome-wide measurements of mRNA stability in E. coli and S. cerevisiae have contradicted the mechanistic expectation that half-life is correlated with mRNA levels. These results imply that natural selection is operating on mRNA levels and/or half-lives, independent of selection on protein levels, and suggest that we analyze the selection pressure on mRNA metabolism to better interpret data on mRNA levels and half-lives. Using an optimal growth assumption, we show that high rates of mRNA turnover can be favored by reduced time to adapt to changing conditions and by reduced stochastic fluctuations in protein levels. Only the adaptation explanation fits the data. The tradeoff between adaptation time and growth rate may explain protein half-lives. The model also shows that 3 !5 pathways for mRNA degradation, which waste ribosome time, need not cost the organism significant fitness. The long-term stability of the 5 fragment of ompA in E. coli, which degrades 3 !5 , does imply a significant fitness cost.
We also present a general result the selective cost of a percent reduction in enzyme activity (the control coefficient) is equal to the enzyme s optimal percent weight; this generalizes previous evolutionary conclusions from metabolic control analysis and holds with linear flux balance analysis assumptions as well.
Introduction
We give a quantitative model of the fitness of unicellular organisms in terms of their growth rate and the time it takes them to adapt to new conditions. The model includes the kinetics of transcription, translation, and mRNA and protein turnover; stochastic variations in protein levels; and estimates of the fitness costs of transcription, translation, and changes in protein and message levels, including mass investment and energy consumption. While even in E. coli a number of critical parameters are unknown or only known to a poor degree of accuracy, the model nonetheless gives insight into several biological questions: the relationship between expression levels and mRNA half lives, the length of protein half lives versus mRNA half lives, and the directionality of mRNA decay.
(Another challenge is how to interpret correlations between different kinetic parameters, or between kinetic parameters and sequence features. These correlations may reflect the mechanisms by which the cell operates, or they may be due to natural selection. To understand how selection may be operating on these parameters, we have been building fitness-based models of mRNA decay along with our kinetic models.)
Models of Fitness for Uni-cellular Organisms
We will assume that fitness is determined by growth rate rather than by energy efficiency and its correlates, biomass production and dispersal. In the absence of a complete model for the growth of a cell, we can still estimate the effect on growth of changes in various parameters by treating the cell as a network of metabolic reactions. For example, flux balance assumptions combined with the metabolic network and with data on cell composition can be used to determine the relationship between growth and various fluxes (Schilling et al 2000). In this perspective, growth is just the flux through the reactions that create new cell mass.
For our purposes we would like to understand the relation between growth and the levels or speed of various enzymes. We will assume that the total density of dry mass in the cell remains fixed (as in Ehrenberg and Kurland 1984). When the level of one enzyme goes down, the level of other enzymes are assumed to go up. In the absence of detailed knowledge of the cell s regulatory mechanisms, we assume that this compensation happens in such a way as to optimize growth. This may overstate the regulatory capacity of the cell, but arguably selection will ultimately cause the same compensation anway.
As an illustrative example, we will start by considering a simple linear pathway with linear kinetics (no enzyme saturation). The flux through this pathway is a simplified growth rate. If we assume the substrate and product concentrations are fixed, and let the concentration of intermediates varys, Heinrich and Schuster (1998) give the steady state flux as
kg = ( S * " qi P ) / (" (1+qi)/(ki*[Ei]) * qi+1 & qn)
where qi s are the equilibrium constants for reaction i and the ki s are the total reaction rates (both forwards and backwards) for enzyme i. Notice the saturation in flux as ki*[Ei] increases this is because the concentration of its product will build up and ultimately limit its flux. Under the constant density assumption, the optimal levels of these enzymes are given by:
fj/fk = sqrt( (1 + qj)/kj / {(1+qk)/kk} * qi+1 & qk )
where fj is the fraction of cell density taken up by the enzymes. (We are assuming that the mass density of the metabolic intermediates is negligible or constant.) Given "fi = 1, this can be rewritten as
fi = " ln kg / " ln ki
We can derive the same result from the power-law approximation (Fell 1992)
kg = " (ki * [Ei])Ci
where Ci = " ln kg / " ln ki is the control coefficient for enzyme i, if we assume that the control coefficients sum to one. To see why they do sum to 1, consider the effect of raising all enzyme levels by the same factor. All fluxes will increase linearly, so flux balance will be maintained, and intermediate levels will not be affected. (This argument does not apply to regulatory cascades and other situations where the enzymes modify each other.)
We can give an intuitive explanation for this result by considering what happens when an enzyme becomes slower. To maintain the same flux, which means maintaining the same levels of intermediates, we need a correspondingly greater amount of enzyme. Because of the density constraint, there will then be an across-the-board reduction in the level of every enzyme, and in flux, by the amount of this extra enzyme. The %change in growth rate, ln kg, is the same as the extra enzyme mass needed to make up for the change ln ki, which is given by fi * " ln ki. Thus, ln kg / " ln ki = fi.
Finally, we can derive this result from a linear programming model. If we assume that the fluxes are linear in enzyme levels, with excess flux being squandered for excess enzyme, then we can write growth as a system of linear equations:
kg d" Jribosome = Vr * fribosome
kg + kmaintenance d" Jenergy = Ve * fenergy
(kmaintenance represents the concept of maintenance energy Koch 1997)
&
kg = (1 resources that do not aid growth)/(resources required to grow at rate 1)
= fgrowth / (" 1/ki)
(notice that the ki s are now weight adjusted)
Why use a linear model when we know that most pathways do not have rate-limiting steps (Fell 1992)? First, enzymes can become rate limiting, e.g. if the level of the enzyme drops greatly. Second, we can simplify the analysis of the effect of reducing the efficiency of an enzyme by assuming that the enzyme levels compensate so that intermediate levels are unchanged. If we ignore the resources that do not aid growth, we get
" ln kg / " ln kj = Vj/kg * " kg / " kj
= kj/kg * -1/(" 1/ki)2 * -1/kj2
= 1/kj / (" 1/ki) = fj
which matches the MCA result (the term on the left is the definition of Cj). If we consider resources that do not aid growth, then even for intermediate processes such as energy production that support both growth and maintenance processes, we get
" ln kg / " ln kj = fj / fgrowth
With these methods we can also model the cost of changing protein levels. The power-law framework gives
" ln kg / " fk = Ck/fk (1-Ck)/(1-fk) = 0 if fk = Ck (optimum)
"2 ln kg / " fk 2 = Ck/fk2 (1-Ck)/(1-fk)2 = 1/(fk * (1-fk)) at optimum
"2 ln kg / " ln fk 2 = -fk / (1-fk) at optimum
The linear framework gives
" ln kg /"+fj = -1/(" 1/Vi) = kg/fgrowth
" ln kg /"+ln fj = " ln kg /"+ln [Ej] = fj/kg * " kg /"+fj = fj/fgrowth
" ln kg /"-ln fj = " ln kg /"-ln [Ej] = 1,
as with this set of assumptions, step j becomes the rate limiting step
Notice that the cost of enzyme level fluctuations is quite different under the two assumptions. The cost of extra enzyme in the linear framework is consistent with Berg s (1992) result that the fitness cost of synthesizing useless protein is at least the fraction of unnecessary protein synthesis, which we can restate (ignoring differences between proteins in their stability) as " ln kg /"+ fj e" 1.
These approaches lead to various measures of the selection cost of codon bias, which is an aspect of selection pressure on gene expression that has received much attention in the literature. Berg and Silva (1997) give a result with an assumption that the ribosome levels increase to keep the free ribosome concentrations constant, and compute the change in ribosome efficiency due to the codon change. (This leads to the same result as our linear framework.) They then assume that " ln kg /" ln kribosome = 1. This gives " ln kg / " dtcodon = fi/MWi * MWr/tribosome, where tribosome is the time for a ribosome to synthesize protein for another one, the MW s are molecular weights, and i is the gene a slow codon is introduced into.) But our analysis of both models give " ln kg / " ln kriboosome = fr, which is much less than one (e.g. see Bremer and Dennis 1996 review of E. coli growth physiology). Even if we consider other mass investments in protein synthesis (elongation factors and tRNA s), the cost seems very high. The idea that " ln kg /" ln kribosome = 1 has been justified by experiments in E. coli where the level of elongation factor EF-Tu was reduced and growth rate went down linearly (Tubulekas and Hughes 1993). We argue that these experiments show that the control system of the cell cannot compensate for the scale of disruption (around 50% less of the enzyme), but do not show how the cell, or selection, would respond to much smaller changes in kr.
Another model by Bulmer (1991) gives an even higher cost because it assumes that the level of ribosomes won t adjust, so that the rate of synthesis of all proteins will drop because of ribosomes stalled at the slow codon, and that kg will drop linearly with the drop in protein synthesis.
Comparison of theoretical costs with experimental data
We can compare two of the theoretical predictions above with data. First, both metabolic control analysis and the flux balance analysis-inspired linear programming model predict that the control coefficient on growth should match the fraction of cell protein:
" ln kg / " ln Vj = fj
The experimental data for the impact of reduced protein levels or activity on growth rates seems consistent with a metabolic control analysis perspective, with a power-law approximation for the effect of enzyme activity changes on flux (Fell 1992). However the control coefficient for the lac permease is very high (over .5). (That is, increasing the level of permease by 10% would increase the growth rate by over 5%.) Our analysis of optimal protein levels under a power-law approximation is that the percent weight of protein should match the control coefficient. The control coefficient for beta-galactosidase of .018 is consistent with this claim, but the control coefficient for permease is obviously way too high. (The beta-galactosidase measurements were actually with reduced enzyme activity mutants not with changed enzyme levels should explain. Another issue with all these experiments is whether the control system of the cell can adapt to the optimal solution our formulas find and that evolution would presumably find if our formulas were correct.) Our explanation is that the phenomenon of lactose killing, which is apparently mediated by the permease (Koch 1983), results in counter-selection for low and relatively limiting levels of permease.
Another issue in the cost of reducing the level of a protein is redundancy. For example, 20-40% yeast proteins show no measurable growth defect when knocked out, even when several media are assayed (Smith et al 1997, Thatcher et al 1998, Winzeler et al 1999). Although this does not mean that there is no growth defect, and it is likely that some of these genes are important only in specific conditions, these results does suggest that for many genes there are mechanisms that reduce the cost of large fluctuations in levels, besides the linear pathway effects discussed above. Analysis of essentiality of duplicated versus singleton genes in yeast suggests that indirect functional redundancy, rather than straightforward duplicated genes, plays a major role (Wagner 2000).
The experimental data for the impact of unnecessary protein on growth rates is difficult to interpret because of artificial evolution during the experiments and because some proteins, such as the lac permease, have deleterious rather than neutral effects (Koch 1983). The experimental effect found by Koch is surprisingly small: " ln kg = .15%, yet fwaste = 2-3%. The effect size is clearly inconsistent with the " ln kg /"+ fj = 1 model described above. Because the unnecessary protein (beta-galactosidase) was not at all useful (no lactose in the medium) we cannot use the metabolic control analysis approach (which finds much smaller costs) to avoid this discrepancy. The negative effects of permease expression found by Koch bring into question the results of previous studies on fitness costs of expressing the lac operon. (They also point out a significant limitation with our models!) Other studies comparing Trp auxotrophs to wild-type E. coli (reviewed by Diamond 1986), but we argue that the Trp operon in the wild-type cells is not truly useless (importing tryptophan into the cell has a cost too). Perhaps more importantly, the effect depends on growing the strains together, suggesting that higher expression of tryptophan importers by is the actual advantage of the auxotroph. Another complication is that the interpretation of most of these studies has been on the basis of the energetic costs of enzyme synthesis or enzyme action (e.g. Koch 1983, Diamond 1986). Units of energy or of glucose consumption provide a currency to unify these costs, but we do not believe that this metric is warranted (these experiments compared growth rates not energy usage or efficiency of biomass production).
Due to all of these experimental and theoretical uncertainties, we will use both the metabolic control analysis models and the linear models to analyze mRNA metabolism.
Explanations for mRNA decay
Messages with longer half-lives will accumulate to higher levels, so naively we would expect highly expressed messages to have longer half-lives. This expectation fails for both E. coli (Bernstein et al 2002) and S. cerevisiae (Wang et al 2002), and for E. coli there is a clear correlation in the other direction. This result also contradicts the most obvious model of selection, where the cost of mRNA turnover (in energy consumption and in mass investment in RNA polymerase) is proportional to message levels. Why do unicellular organisms spend significant resources to turn over their mRNA so rapidly? We consider two potential benefits of mRNA turnover: to reduce stochastic fluctuations in protein levels and to reduce the time for the organism to adapt to new conditions.
At steady state, we have
[M] = kt/(mdeg + kg)
[P] = ktot * [M] / (pdeg + kg) = kt * ktot / { (mdeg + kg) * (pdeg + kg) }
m = ln(2)/mdeg
p = ln(2)/pdeg
Given a fixed [P] and kg, we have four parameters (kt, ktot, mdeg, pdeg) and three degrees of freedom. If we consider [M] fixed as well, then there is one degree of freedom between mdeg and kt and another between ktot and pdeg.
Cost of mRNA decay
We can analyze the cost of mRNA decay by measuring the cost in RNA polymerase mass and energy for the increased mRNA synthesis to balance the decay. (This implicitly assumes that mRNA degradation is free in both energy and mass cost for enzymes, but we expect both to be far lower than RNA polymerase costs.) By focusing on just the RNAP activity that goes to counteract mRNA decay, and the associated energy costs, and using the linear framework, we see that
" ln kg / " ln mdeg = (fRNAP + fE * %energy for counteracting mRNA decay)
= (fRNAP + eRNAP * fE) * mdeg/(mdeg+kg)
" ln kg / " mdeg = " ln kg / " ln mdeg / mdeg = (fRNAP + eRNAP * fE)/(mdeg+kg)
Please note that the non-linear selective cost of mdeg is entirely an illusion as the top will scale with mdeg+kg as well. Also note that this model assumes that all genes mRNA half-lives change in unison. For changes to a single gene we get:
" ln kg / " mdeg(i) = fi * (fRNAP + eRNAP * fE)/(mdeg+kg)
The RNAP mass cost per unit mdeg(i) can also be written as MWrnap * tRNAP(i) * [Mi], where tRNAP(i) is the time for the polymerase to synthesize one complete message and the concentration of message [Mi] is expressed in moles per standard weight of dry mass.
Also note that we can get similar results from the metabolic control analysis framework we can think of message decay as reducing the efficiency of RNA polymerase, or we can think about compensating changes to RNA polymerase and energy-producing enzyme levels to maintain the same levels of free polymerase and ATP (so that the rate of synthesis of other messages is unaffected). This relation is given by
" [RNAP] / " mdeg(i) = tRNAP(i) * [Mi]
" fRNAP / " mdeg(i) = MWRNAP * tRNAP(i) * [Mi]
which leads to the same result.
Stochastic Benefit of mRNA decay
The benefit of turnover arises because the variance in a protein s expression is a multiple of both the protein levels and the number of polypeptides produced per transcript (Berg 1978; Thattai and van Oudenaarden 2001; Ozbudak et al 2002). The number of polypeptides produced per transcript = ktot/mdeg, giving
Var(Pi) =Pi*(1 + ktot(i)/mdeg(i))(1+Hi)*(1+pdeg(i)/mdeg(i))
where Pi is the number of copies of the enzyme per cell and Hi is a constant representing the strength of transcriptional negative feedback on fluctuations in this protein s levels. For simplicity, we will assume pdeg(i) << mdeg(i) in this analysis.
The cost of these fluctuations in the MCA analysis is given by
" ln kg = * Var(fi) * "2 ln kg / "fi2
= * f1i2 * Var(P) /(fi *(1-fi))
= * f1i2 * Pi * 1/(fi*(1-fi)) * (1 + ktot(i)/mdeg(i))/(1+Hi)
= * f1i /(1-f i) * (1 + ktot(i)/mdeg(i))/(1+H i)
where f1i is the %weight per cell of one copy of the protein, so that f i = f1i *Pi. We assume that the second-order approximation and the underlying power-law approximation are both valid for the size of fluctuations, but arguably large fluctuations for essential proteins are implausible because they would prevent cells from growing entirely. As the cell gets smaller, f1i rises and the cost of fluctuations increases. More surprising is the very weak dependence of the cost on the level of the protein (fi << 1 implies the denominator has little effect), and the increased effect with heavier proteins.
At the optimal balance between costs and benefits,
"cost ln kg / " mdeg(i) = "stochastic ln kg / " mdeg(i)
MWRNAP * tRNAP(i) * [Mi] = * f1i /(1-f i) * ktot(i) / (mdeg(i)2 * (1+Hi))
mdeg(i)2 = * f1i /(1-f i) * ktot(i) / (MWRNAP * tRNAP(i) * [Mi] * (1+Hi))
now [Mi] = [Pi] * (kg+pdeg(i))/ktot(i)
[Pi] = fi / MWi
1/[Mi] = MWi/fi * ktot(i)/(kg+pdeg(i))
mdeg(i)2 = * f1i/(fi*(1-f i)) * MWi * ktot(i)2/(MWRNAP * tRNAP(i) * (kg+pdeg(i))*(1+Hi))
H" * f1*MWi/(fi*(1-f i)) * ktot(i)2/(MWRNAP*tRNAP1*(kg+pdeg(i))*(1+Hi))
mdeg(i) = ktot(i) * " f1*MWi /(2*(fi*(1-f i))*MWRNAP*tRNAP1* (kg+pdeg(i)) * (1+Hi))
where f1 is the %weight of one amino acid and tRNAP1 is the time to synthesize one codon (and ignoring time to synthesize introns and untranslated regions).
In the linear analysis we get
" ln kg = "Var(fi) /fi * (" kg /"+ln fi + " kg /"-ln fi)/2
= "Var(Pi)/Pi * (1+fi)/2
= * (1+fi) * "(1 + ktot(i)/mdeg(i))/((1+Hi)*Pi)
= * (1/"fi + "fi) * "f1i*(1 + ktot(i)/mdeg(i))/(1+Hi)
Note that the 1/"fi dominates as "fi << 1. The optimal balance is given by
"cost ln kg / " mdeg(i) = "stochastic ln kg / " mdeg(i)
MWRNAP * tRNAP(i) * [Mi] = * 1/m deg(i)2
* (1/"fi + "fi) * "f1i/((1 + ktot(i)/mdeg(i))*(1+Hi))
H" * 1/m deg(i)2 * " f1i* mdeg(i) / (ktot(i) * (1+Hi) * fi)
mdeg(i)3/2 = * MWi/fi * ktot(i)/(tRNAP(i)*(kg+pdeg)) * " f1i / (ktot(i) * (1+Hi) * fi)
= * MWi/(fi3/2 * tRNAP(i) * (kg+pdeg)) * "f1i* ktot(i) /(1+Hi)
The approximation is justified because the number of proteins synthesized per message is typically much greater than 1 (e.g. Bremer and Dennis 1996).
The main point to notice with both models is that the decay rate decreases as the abundance of the protein goes up. The MCA analysis gives a square root dependence while the linear model gives a linear dependence.
Adaptation
Simple model of switching
Math: Tup = ln(2)/mdeg + ln(2)/pdeg, d(ln kg) = d(Tup) / horizon
Ensemble concept
Complete model of switching
% change in ensemble , constant Psynth, issues around [RNAP]
Tradeoff ! horizon ~ few hours???
In contrast, the adaptation time explanation appears to fit the data for message half-lives. If we assume that both transcription rates and turnover rates are similar during adaptation and during steady growth, the adaptation time (time until the new protein reaches steady state levels) is linear in the message and protein half-lives (Hargrove et al 1991). With a simple model of growth and regulation, where the organism spends a certain time in a given condition (which we will call the time horizon) and then switches to a new condition (Savageau 1998), the reduction in fitness due to adaptation time is the fraction of the time horizon taken up by adaptation. (We ignore the cost of continuing to synthesize obsolete proteins, as the cost of synthesizing useless protein is small relative to the cost of not being able to grow at all due to lack of an essential protein.) Analyzed one gene at a time, this again leads to higher half-lives for more highly expressed genes, but adaptation requires an ensemble of proteins. Hence messages for proteins with related functions, which need to switch on in concert, should have similar half-lives, which is in fact true for E. coli (Bernstein et al 2002) and S. cerevisiae (Wang et al 2002). And the relationship between message half-life and message levels disappears proteins which are turned on in concert may be expressed at very different levels. With cost parameters which seem consistent with physiological data (Bremer and Dennis 1996; Koch 1983), the typical E. coli message half-life of 4 minutes corresponds to a time horizon (time between switches) of around 100 minutes (XXX). This is consistent with the ecology of E. coli: for example, in an adult human host, E. coli will spend several hours in the lactose-rich upper small intestine followed by several hours in the maltose-rich lower small intestine (Savageau 1998). Our model of adaptation time does not explain the modest correlation between high expression and short mRNA half-lives in E. coli (Bernstein et al 2002), but in principle this could be accounted for if genes needed at high levels at high growth rates (where the measurements were taken) tend to have a shorter time horizon.
Horizon and protein decay
Stringent response complicates this all greatly?
This adaptation time model may also help explain protein turnover rates, which are much lower than mRNA turnover rates but are still a significant drag on steady state growth. The cost of turning over protein is far higher than the cost of turning over message (RNA polymerase is a fraction of the size of a ribosome, and many proteins are synthesized for each message), which explains why the protein half-lives in E. coli are perhaps ten times longer than message half-lives.
Directionality of mRNA decay
The growth optimization framework can formalize an old debate about the directionality of mRNA decay. 5 !3 decay can proceed without wasting ribosome time, while 3 !5 decay must waste ribosome time, and early research assumed that directional decay was 5 !3 (Kennell and Riezman 1977). Nevertheless, some messages in E. coli, including highly expressed ribosomal protein messages, clearly decay by a 3 !5 pathway (Coburn and Mackie 1999). Our analysis shows that the fitness cost of ribosome time wasted by a 3 !5 decay can be surprisingly modest, especially for short ORFs. Because of the large number of polypeptides synthesized per transcript, and because there are relatively few ribosomes on a transcript at a given time (from parameters in Bremer and Dennis 1996), the loss of ribosome time due to a single slow codon may be close to as large as the loss of ribosome time represented by the ribosomes that are already on the message when its 3 end is degraded.
The cost of 3 !5 decay is very high if the truncated message is stable, however, because the intact 5 end will continue to initiate translation and, with a truncated 3 end, all of the ribosome time is wasted. Yet there is a major message in E. coli, ompA, that decays 3 !5 and whose 5 end is much more stable than the 3 end, by up to 10 minutes (von Gabain et al 1983). We speculate that the truncated ompA message is sequestered in a ribosome-free state.
Optimal translation rates
We are in the process of analyzing the tradeoff between ribosome queuing at high translation rates (Liljenstrom and Blomberg 1987) and mass investment in message at low translation rates. We suspect that this may explain the known correlation (Sakai et al 2001) between translation initiation features such as a strong Shine-Dalgarno sequence and AUG initiation codon, and codon bias, which is characteristic of highly expressed genes.
Discussion / Limitations
Operons
Other problems with fitness theories?
Need for compensating changes & limit of selection
Detailed comparisons of data?
Ensemble concept + regulation make this seem implausible&
Can at least do more organisms
Conclusions
In conclusion, fitness models that combine growth rates and adaptation times can explain the half-lives of mRNA and perhaps protein in unicellular organisms, while stochastic fluctuations are not consistent with the observed pattern. Apparently, the cell s control mechanisms suffice to reduce fluctuations without requiring mRNA turnover. Our fitness model also shows that the cost of degrading highly expressed mRNAs in E. coli by the 3 !5 is actually rather small. Models of fitness that combine adaptation time and growth rates may also have broader relevance. For example, the design of signaling pathways must include tradeoffs between the mass investment and energy consumption costs of the pathway at steady state and the response time and reliability of receiving the signal. The tradeoffs should also apply to multicellular organisms: although a fitness function based on cellular growth rates is clearly inappropriate, energy consumption and mass investment still have costs.
References
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