The fitness landscapes of translation
TThe fitness landscapes of translation
Mario Josupeit and Joachim Krug
Institute for Biological Physics, University of Cologne, Z¨ulpicher Strasse 77, 50733 K¨oln, Germany
Abstract
Motivated by recent experiments on an antibiotic resistance gene, we investigate ge-netic interactions between synonymous mutations in the framework of exclusion modelsof translation. We show that the range of possible interactions is markedly differentdepending on whether translation efficiency is assumed to be proportional to ribosomecurrent or ribosome speed. In the first case every mutational effect has a definite signthat is independent of genetic background, whereas in the second case the effect-sign canvary depending on the presence of other mutations. The latter result is demonstratedusing configurations of multiple translational bottlenecks induced by slow codons.
Keywords: translation, fitness landscape, synonymous mutations, exclusion process
1. Introduction
All living cells synthesize proteins by transcribing the hereditary information in theirDNA into strands of messenger RNA (mRNA) which are subsequently translated intoamino acid sequences. The genetic code that assigns triplets of nucleotides (codons)to their corresponding amino acids is redundant, since most amino acids are encodedby several codons. Mutations that change the DNA sequence (and hence the sequenceof codons) but leave the amino acid sequence unchanged are called synonymous. Suchmutations were long thought to have no phenotypic consequences and hence to be evolu-tionarily silent. However, meanwhile many cases have been reported where synonymousmutations profoundly affect organismal functions, primarily by modifying the efficiency,timing and quality of protein production [1, 2, 3, 4, 5].Our work was motivated by a recent experimental study of synonymous mutationsin a bacterial antibiotic resistance enzyme that inactivates a class of drugs known as β -lactams. A panel of 10 synonymous point mutations was identified which significantlyincrease the resistance against cefotaxime, as quantified by the drug concentration atwhich a fraction of 10 − of bacterial colonies survives [6]. Moreover, several of these mu-tations display strong nonlinear interactions in their effect on resistance [7]. Of particularinterest are interactions referred to as sign-epistatic [8]. In the present context this im-plies that a mutation that increases resistance in the genetic background of the originalenzyme, and hence has a beneficial effect on bacterial survival, becomes deleterious inthe presence of another mutation, or vice versa. Sign epistasis is a key determinant ofthe structure of the fitness landscape of an evolving population [9]. Preprint submitted to Physica A September 23, 2020 a r X i v : . [ q - b i o . S C ] S e p ere we explore possible mechanisms that could explain sign epistatic interactionsbetween synonymous mutations. In line with the experimental observations [7], we as-sume that the effect of the synonymous mutations on organismal fitness is mediatedby the efficiency of protein translation. The process of translation can be described bystochastic particle models of exclusion type, which have been used in the field for morethan 50 years [10, 11, 12, 13, 14, 15]. These models treat ribosomes as particles movingunidirectionally along a one-dimensional lattice of sites representing the codons. Theexclusion interaction ensuring that two particles cannot occupy the same site accountsfor ribosome queuing [15, 16].Within this conceptual framework we argue that the occurrence of sign epistatic in-teractions depends crucially on the definition of translation efficiency. If efficiency isidentified with the stationary ribosome current, we prove rigorously that sign epistasisis not possible. In contrast, if efficiency is taken to be the average speed of a ribosome(equivalently, the inverse of the ribosome travel time), then plausible scenarios givingrise to sign-epistatic interactions are readily found, and fitness landscapes that are qual-itatively similar to those observed experimentally in [7] can be constructed [17]. Weconclude, therefore, that there are (at least) two different fitness landscapes of transla-tion, one of which is always simple, whereas the other displays a complex structure ofhierarchically organized neutral networks.
2. Inhomogeneous TASEP
The simplest model of translation is the totally asymmetric simple exclusion process(TASEP) illustrated in Fig. 1. The mRNA strand is represented by a one-dimensionallattice of length L and ribosomes are particles occupying single lattice sites. Particlesenter the lattice at the initiation rate α and leave the lattice at the termination rate β .Within the lattice particles move from site i to site i + 1 at the elongation rate ω i with1 ≤ i ≤ L −
1. This version of the TASEP neglects a number of important featuresof translation, such as the spatial extension of ribosomes and the stepping cycle of theelongation process, which have been included in more refined variants of the model[18, 19, 20, 21]. Here we restrict ourselves to the simplest setting, as we expect theconceptual points that we wish to make to be robust with respect to such modifications. α ω ω β Figure 1: Schematic of the inhomogeneous TASEP on a lattice of L = 10 sites. Particles enter thesystem at rate α , hop from site i to site i + 1 at rate ω i and exit at rate β . Backward jumps and jumpsto occupied sites are forbidden. Importantly, the elongation rate ω i depends on the identity of the codon associatedwith the site i , which implies that the TASEP is inhomogeneous with sitewise disorder[22]. Whereas the stationary state of the homogeneous TASEP with constant elongationrates ω i ≡ ω is known exactly [23, 24], only approximate approaches are available forthe inhomogeneous model [25, 26, 27, 28]. Key observables of interest in the present2ontext are the stationary particle current J , which is site-independent because of massconservation, the site-dependent mean occupation numbers ρ i , and the spatially averagedmean particle density ¯ ρ = 1 L L (cid:88) i =1 ρ i . (1)Additionally we consider the mean travel time required for a particle to move across thelattice, which is given by [29] T = L ¯ ρJ = Lv = Lτ. (2)Here v = J/ ¯ ρ denotes the average particle speed and τ = v the mean elongation timeper site.
3. Translation efficiency
In order to quantify the effects of synonymous mutations on protein expression andfitness, we need to link the TASEP observables defined above to the efficiency of trans-lation. The most commonly used efficiency measure is the rate of protein productionper mRNA transcript, which (assuming steady-state conditions) corresponds to the sta-tionary particle current J [30]. In experiments this quantity can be estimated as theratio between the cellular abundance of a protein and that of the corresponding mRNA[31]. In ribosome profiling experiments, which take snapshots of the ribosome occupancyalong the transcript, efficiency is instead associated with the fraction of codons that arecovered by ribosomes, corresponding to the mean particle density ¯ ρ in the TASEP [32].In the absence of ribosome queuing these two measures would be expected to be propor-tional to each other, but empirically they are found to correlate poorly [1], indicatingthat queuing cannot generally be ignored.The efficiency measures defined so far assume implicitly that ribosomes are sufficientlyabundant, such that translation initiation occurs readily as soon as the initiation site isfree. However, the production of ribosomes is very costly for the cell, and it has thereforebeen suggested that translation is optimized towards using each ribosome as efficientlyas possible [33, 34, 35, 36]. Within the TASEP setting this implies that the relevantquantity associated with translation efficiency is the ribosome travel time T as a measureof translation cost [37, 38]. Equivalently, efficiency is quantified by the average speed v of the ribosome. In the following we examine the response of the ribosome current J and the ribosome speed v to synonymous mutations, and argue that v can displaysign-epistatic interactions whereas J cannot. The stationary current in the inhomogeneous TASEP is a (generally unknown) func-tion J ( α, β, ω , . . . , ω L − ) of the rates. In the Appendix we establish rigorously that thisfunction is monotonic in its arguments. This implies that a synonymous mutation thatreplaces one of the elongation rates ω i by a rate ˜ ω i always increases the ribosome currentif ˜ ω i > ω i and decreases it if ˜ ω i < ω i , irrespective of the values of the other rates. As aconsequence, the effect-sign of any synonymous mutation is independent of the geneticbackground, and sign epistasis cannot occur.3 numerical study of the inhomogeneous TASEP reported results for the stationarycurrent that appear to contradict the monotonicity property [39]. The authors considereda binary system where a fraction f of the sites are assigned a slow rate ω i = p < − f have rate ω i = 1. For certain conditions they observed that thestationary current was increasing with increasing f , or decreasing with increasing p . Wehave repeated these simulations and find the expected monotonic behavior throughout[17]. The results reported in [39] may have been caused by insufficient relaxation to thestationary state . We base the discussion of the ribosome speed on a simple rate configuration witha single slow site (the ‘bottleneck’) with rate ω k = b < ω i = 1 for i (cid:54) = k . The TASEP with a bottleneck has been the subjectof numerous studies [40, 41, 42], and although its stationary state is not known exactly,many features of the model are well understood. For convenience we focus on the casewhere the initiation and termination rates are large, α = β = 1, such that the behaviorof the system is dominated by the bottleneck. In the limit of large L the system thenphase separates into a high density region (the ‘traffic jam’) preceding the bottleneck anda low density region behind it. Continuity of the current implies that the densities of thetwo regions are related by (cid:37) ≡ ρ low = 1 − ρ high . The density (cid:37) is determined through therelation (cid:37) (1 − (cid:37) ) = j ( b ), where j ( b ) denotes the maximal current that can flow throughthe bottleneck site. The function j ( b ) is not explicitly known, but it has been establishedthat j ( b ) < j (1) = for any b < j ( b ) = b (1+ b ) , and more accurate power series expansions can befound in [27] and [40]. Importantly, (cid:37) ∈ (0 , ) is uniquely determined by b , and we maytherefore use (cid:37) to quantify the strength of the bottleneck.Whereas the current is fully determined by the bottleneck strength, the ribosometravel time and speed also depend on its location k . As we are generally concerned withthe limit of large L , it is useful to introduce the scaled position x = k/L for k, L → ∞ .Then the average density in the stationary state is given by¯ ρ ( (cid:37), x ) = x (1 − (cid:37) ) + (1 − x ) (cid:37) (3)and correspondingly the mean elongation time is τ = ¯ ρ(cid:37) (1 − (cid:37) ) = x(cid:37) + 1 − x − (cid:37) = 11 − (cid:37) + x − (cid:37)(cid:37) (1 − (cid:37) ) . (4)Since (cid:37) < , the elongation time is an increasing function of x . It is straightforward toshow that for x < (cid:37) the elongation time is smaller than the value τ = 2 in the absence ofa bottleneck. Thus by placing the bottleneck sufficiently close to the initiation site theribosome travel time can actually be reduced compared to the homogeneous system [38].This effect has been invoked to explain the evolutionary benefits of the accumulation ofslow codons at the beginning of mRNA transcripts that is observed in genomic data [37]. M.E. Foulaadvand, private communication. FF F F I 0 1 2 FF F F III 0 1 2 FF F F V0 1 2 FF F F II 0 1 2 FF F F IV 0 1 2 FF F F VI Figure 2: All possible fitness landscapes for a system with two bottlenecks. The plots show the fitnessvalues of the four genotypes as a function of the number of mutations. In all cases the fitness of thedouble mutant is equal to F . If fitness is associated with the ribosome current the landscapes has to beof type I. Landscapes V and VI display instances of sign epistasis which are marked in red.
4. Multiple bottlenecks
Consider now a situation where synonymous point mutations at two sites k and k ofthe mRNA replace regular codons with elongation rate ω i = 1 by bottlenecks with rates b , b <
1. The scaled positions of the two sites are denoted by x ν = k ν /L , ν = 1 ,
2, andwe assume without loss of generality that b < b . This defines a simple genetic systemcomprising four genotypes, the unmutated sequence (0), the single mutants ν = 1 , j ( b ) < j ( b ), bottleneck 2 is irrelevantin the presence of bottleneck 1. As a consequence, the double mutant is phenotypicallyindistinguishable from the system containing only bottleneck 1. We say that mutation 2is conditionally neutral in the presence of mutation 1.The fitness landscape spanned by the four genotypes thus contains 3 distinct fitnesslevels that we denote by F , F = F and F . If fitness is associated with the ribosomecurrent the ordering of the fitness values is dictated by the strength of the bottlenecksand given by F > F > F = F . By contrast, through a suitable choice of thebottleneck positions, for the elongation time or the ribosome speed all 3! = 6 possibleorderings of fitness values can be realized. The resulting fitness landscapes are illustratedin Fig. 2. In 2 of the 6 cases sign epistasis occurs, in that the addition of bottleneck 1 caneither increase or decrease the ribosome speed depending on the presence of bottleneck 2.5 x x (a) IVIIVII V III % = x τ = τ % = x x x (b) Figure 3: (a) Predicted phase diagram in the plane of scaled bottleneck positions ( x , x ) displayingthe regions where the orderings I-VI depicted in Fig. 2 are realized for the ribosome speed. Along thevertical line x = (cid:37) the travel time of the system with bottleneck 1 is equal to the travel time of thesystem without bottlenecks, τ = τ = 2, and correspondingly along the horizontal line x = (cid:37) thecondition τ = 2 holds. The slanted line is determined by the condition τ = τ . In the shaded regionsV and VI sign epistasis is present. (b) Numerical verification of the predicted phase diagram based onsimulations of a TASEP with L = 800. The stationary current J and density ¯ ρ were determined for thetwo bottlenecked systems by averaging over 5 × time steps following an equilibration period of 10 time steps. The travel times were computed from (2), and compared to the travel time τ of the systemwithout bottlenecks obtained from a separate simulation. Symbols show the ordering of the simulatedtravel times τ , τ and τ . The bottleneck densities chosen for this image are (cid:37) = 0 . (cid:37) = 0 . Using the expression (4) for the elongation time, the different orderings can be mappedto regions in the plane of scaled bottleneck positions ( x , x ) ∈ [0 , . Because of thelinearity of (4) these regions are delimited by straight lines (Fig. 3). Figure 3(b) shows aphase diagram obtained from simulations of finite systems, which agrees very well withthe prediction based on the asymptotic behavior for L → ∞ .These considerations generalize straightforwardly to systems with N > N genotypes in total. The phenotype (ribosomecurrent or ribosome speed) of a given genotype is determined by the strongest bottle-neck that is present in the system. Labeling the bottlenecks in decreasing order of theirstrengths, b < b < . . . < b N <
1, it follows that all 2 N − genotypes in which bot-tleneck 1 is present share the same fitness value F . Among the 2 N − genotypes thatlack bottleneck 1, half are dominated by bottleneck 2, and so on. This leads to a hier-archical structure of neutral regions that is illustrated in Fig. 4 for N = 4. If fitness istaken to be proportional to the ribosome current, the N + 1 fitness values are orderedas F < F < . . . < F N < F , whereas all possible ( N + 1)! orderings can be realized forthe ribosome speed.
5. Conclusion
In this article we have outlined a possible scenario for the occurrence of sign-epistaticinteractions between the effects of synonymous mutations on the efficiency of protein6 F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F Figure 4: (a) The 2 = 16 genotypes in a system with 4 bottlenecks form a 4-dimensional hypercube,where neighboring nodes are connected by the addition or removal of a bottleneck. The indices of thefitness values show which bottlenecks are present in the genotype. (b) Under the model the hybercubedecomposes into N − K genotypes each ( K = N − , N − , . . . ,
1) on which thefitness is constant and determined by the strongest bottleneck that is present. In addition there are twosingle nodes corresponding to the system without a bottleneck ( F ) and the weakest bottleneck ( F N ).In the figure the subcubes of size 8, 4 and 2 are marked in color. production. A necessary requirement for the scenario to apply is that translation effi-ciency is related to ribosome speed rather than to ribosome current. We have workedout a detailed quantitative description for the idealized situation of a mRNA transcriptwith homogeneous elongation rates into which a small number of well-separated slowcodons (“bottlenecks”) are inserted, but we expect our considerations to generalize tomore realistic elongation rate profiles. The key feature of the translation process thatenables sign epistasis is the fact that, somewhat counter-intuitively, the introduction ofslow codons can decrease the ribosome travel time by reducing the amount of ribosomequeuing [36, 37, 38]. To the extent that ribosome queuing is a generic feature of trans-lation [15, 16], sign-epistatic interactions in the ribosome speed are therefore likely toarise.The model of discrete, well-separated bottlenecks predicts a specific fitness landscapestructure where the hypercube of genotypes decomposes into lower-dimensional subcubesof constant fitness (Fig. 4). The combination of strong sign-epistatic interactions withextended plateaux of approximately constant resistance levels is indeed a visually strik-ing feature of the experimental data set that motivated this work [7]. A quantitativecomparison between the model and the data is however beyond the scope of this articleand will be presented elsewhere. Acknowledgments
JK is grateful to Martin Evans, Grzegorz Kudla, Mamen Romano and Juraj Szavits-Nossan for helpful discussions, and to SUPA and the Higgs Centre for Theoretical Physicsin Edinburgh for their gracious hospitality during the early stages of the project. Thework was supported by DFG within CRC 1310
Predictability in Evolution . We dedicate7his article to the memory of Dietrich Stauffer, fearless explorer of disciplinary boundariesand translator of scientific and cultural idioms.
Appendix: Proof of the monotonicity property
We prove that the stationary current in the inhomogeneous TASEP with open bound-aries is a monotonic function of the jump rates. The proof is based on the waiting timerepresentation of the TASEP explained in [44]. Briefly, the TASEP occupation variables σ i ∈ { , } , i = 1 , . . . , L , are mapped to a single-step (SS) interface configuration h ( i )through the relation σ i = 12 [1 + h ( i ) − h ( i + 1)] (5)such that a growth event h ( i ) → h ( i ) + 2 corresponds to the jump of a particle fromsite i − i . Thus the height variables measure the time-integrated local particlecurrent. A dual description of the SS growth process is provided by the waiting timevariables t ( i, j ) denoting the time at which the interface height reaches the point ( i, j )on the underlying tilted square lattice, i.e. when h ( i ) = j (see Fig. 1 of [44] for anillustration of the geometry). The SS/TASEP growth rule implies that t ( i, j ) = η ( i, j ) + max [ t ( i − , j − , t ( i + 1 , j − , (6)where the η ( i, j ) are exponentially distributed random variables. For the open boundaryTASEP on a lattice of L sites the recursion (6) holds for 2 ≤ i ≤ L − i = 1 and i = L . These modifications are notimportant here as the initiation and termination rates α and β will be assumed to befixed.The solution of (6) can be expressed as t ( i, j ) = max π ∈ P i,j T ( π ) (7)where P i,j is the set of upward directed paths π on the tilted square lattice that end at( i, j ), and the passage time T ( π ) of a path is the sum of the random variables along thepath, T ( π ) = (cid:88) ( x,y ) ∈ π η ( x, y ) . (8)The problem of finding the solution of (7) is known as last passage percolation [43, 45].At long times the SS interface attains some stationary velocity c > t →∞ h ( i ) t = c independent of i . Since the height increases by 2 for each jump of aTASEP particle, the velocity is related to the stationary current J through c = 2 J .Correspondingly, by the law of large numbers the rescaled waiting times converge aslim h →∞ t ( i, h ) h = 1 c = 12 J (9)which expresses the stationary current in terms of the last passage percolation problem(7). 8n the inhomogeneous TASEP the distribution of the random waiting times η ( i, j )depends on the site label i . Specifically, the probability density of η ( i, j ) is given by p i ( η ) = ω i e − ω i η . To generate a realization of the TASEP process for a given set of rates { ω i } i =1 ,...,L − , we first draw a set of uniform random variables u ( i, j ) ∈ [0 ,
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