An equilibrium model for ribosome competition
AAn equilibrium model for ribosome competition
Pascal S. Rogalla
Institute for biological and medical engineering, Schools of engineering, biology andmedicine, Universidad Catolica de Chile, ChileDepartment of chemical and bioprocess engineering, School of engineering,Universidad Catolica de Chile, ChileI. Physikalisches Institut (IA), RWTH Aachen University, 52074 Aachen, GermanyE-mail: [email protected]
Timothy J. Rudge
Institute for biological and medical engineering, Schools of engineering, biology andmedicine, Universidad Catolica de Chile, ChileDepartment of chemical and bioprocess engineering, School of engineering,Universidad Catolica de Chile, Chile
Luca Ciandrini
CBS, Universit´e de Montpellier, CNRS and INSERM, Montpellier, FranceLaboratoire Charles Coulomb (L2C), Universit´e de Montpellier and CNRS,Montpellier, FranceE-mail: [email protected]
Abstract.
The number of ribosomes in a cell is considered as limiting, and geneexpression is thus largely determined by their cellular concentration. In this workwe develop a toy model to study the trade-off between the ribosomal supply and thedemand of the translation machinery, dictated by the composition of the transcriptpool. Our equilibrium framework is useful to highlight qualitative behaviours and newmeans of gene expression regulation determined by the fine balance of this trade-off.We also speculate on the possible impact of these mechanisms on cellular physiology.
Keywords : Quantitative Biology, Modelling, mRNA translation, cell physiology a r X i v : . [ q - b i o . S C ] O c t n equilibrium model for ribosome competition
1. Introduction
Gene expression is a costly process which requires the constant involvement ofthe components of its transcriptional and translational machineries (polymerases,transcription factors, ribosomes, tRNAs,...) for protein synthesis. The interplay betweencellular resources and protein synthesis is a topic that has recently drawn the attentionof a growing community (see for instance [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] to mentionjust a few recent publications), and its understanding is pivotal to the developmentof a quantitative characterisation of gene expression regulation. This could also unlockapplications related to, for instance, the design of gene circuits, the control of the cellularphysiology or optimal protein synthesis for biotechnological applications.When modelling transcription and translation, cellular resources such as transcriptionfactors and RNA polymerases are often considered as being unlimited, with just a fewrecent exceptions [1, 11, 12]. At the translational level, ribosomes are known to belimiting [13] and their abundance is strongly related to expression levels and cellularphysiology [14, 15, 16]. However, even state-of-the-art tools such as the RibosomeBinding Site (RBS) calculator [17], which is able to predict the mRNA’s recruitmentrate of a ribosome given the nucleotide sequence of the transcript, neglect the possibledepletion effects that strong synthetic RBSs might have on the ribosomal pool, and thuspotentially induce protein burden.For all these reasons here we develop a toy model to study the trade-off betweendemand (mRNA abundances) and supply of resources (ribosomes). Although neglectingmany other important aspects of resource competition and cell physiology -for instancemetabolism-, we show that even this simple coarse grained model predicts a richbehaviour and new means of gene expression regulation just by considering ribosometrade-off between different mRNA populations. Furthermore, our model provides anintuitive mechanistic explanation of the phenomenological growth law relating ribosomeabundances and growth rates.In the next section we first revise an equilibrium model of gene expression that wetake as a reference point for our analysis, then generalise it to adapt it to the trans-lation process in which several ribosomes can concurrently translate an mRNA. In thelast part of the paper we speculate on how the model predictions can be interpreted torationalise gene expression regulation by the trade-off between supply and demand ofribosomal resources.
2. Model
In this section we summarise the basics of a thermodynamic model of gene expressionbased on the assumption that transcription occurs when an RNA polymerase (RNAP)is bound to a promoter of interest [18, 19]. Equilibrium statistical mechanics provides n equilibrium model for ribosome competition p that an RNAP is bound to a promoter, and geneexpression levels are assumed to be proportional to that quantity. In more detail, if m is the concentration of mRNAs and k TX their transcription rate, one can write dmdt = k TX p − γm , (1)where γ is the degradation rate of mRNAs. At the steady-state one then finds that m = p k TX /γ , which can be used as a proxy for gene expression.Interactions with cis- or trans- regulatory elements will not be considered heresince they are not related to the topic of our work, and detailed reviews can be foundin [20, 21]. Instead, here we focus only on the role of particle (RNAP, ribosome) com-petition, and show how relative expression between different genes could arise purely bytrade-off between those elements.To introduce the model we follow the procedure established in [18] that considersa single specific site (playing the role of a promoter and characterised by an occupationnumber n b ) immersed in a pool of n particles (RNAPs) and N non-specific sites (DNAsites other than the promoter of interest). The specific site can be occupied by aparticle ( n b = 1) or be empty ( n b = 0), and the remaining n f = n − n b particles aredistributed in the N non-specific sites. Since the expression rate of the gene of interest isdirectly related to the occupation of the specific site, we need to compute the probability p ( n b = 1) of finding the particle bound specifically. For the sake of completeness, belowwe re-derive the results of this model, but we refer for instance to [18, 20, 21] for moredetailed explanations.According to equilibrium statistical mechanics, the probability of the macrostate j withfree energy E ( j ) is given by the Boltzmann distribution p ( j ) = D ( j ) e − βE ( j ) /Z tot , with β = 1 /k B T being the inverse temperature and D ( j ) the degeneracy of the macrostate-the number of microstates with the same energy. The sum of all partition functionsof the macrostates is the total partition function Z tot , which gives the normalisationfactor of the probability so that (cid:80) j p ( j ) = 1. By identifying the macrostate with theoccupation number n b we are then able to compute p ( n b = 1) knowing (cid:15) b := E ( n b = 1)and (cid:15) f := E ( n b = 0).In order to simplify the mathematical expressions, the partition functions Z ( j )and the degeneracies D ( j ) are usually written in terms of a reference state, here n b = 0, and indicated with the subscript 0: Z ( n b ) := Z ( n b ) /Z ( n b = 0) and D ( n b ) := D ( n b ) /D ( n b = 0). The expression for p ( n b = 1) can therefore be writtenas p ( n b = 1) = Z ( n b = 1) Z tot0 = D ( n b = 1) e − β ∆ (cid:15) D ( n b = 1) e − β ∆ (cid:15) , (2)where ∆ (cid:15) := (cid:15) b − (cid:15) f stands for the energy turnover by the process of binding theparticle.The degeneracy D ( j ) counts the possible ways to arrange the non-specifically boundRNAP to the non-specific sites in the case of state j . Here, the non-specific sites are n equilibrium model for ribosome competition D ( n b = 1) (cid:39) n/N (assuming N (cid:29) n ), that can beplugged in Eq. (2) to obtain p ( n b = 1) = nN e − β ∆ (cid:15) nN e − β ∆ (cid:15) . (3) Our model is a direct extension of this equilibrium framework. Since we present it interms of mRNA translation, we will consider ribosomes instead of RNAPs, landing onribosome binding sites or 5’UTRs (untranslated regions) instead of promoters. However,we keep the same notations because of the generality of the mathematical structure un-derlying the results, that could then immediately re-interpreted in other systems (aswell as transcription).The mRNA pool is considered to be in equilibrium with a reservoir of n f free ribo-somes diffusing in the cytoplasm. The volume available to the ribosomes in the reservoirthen can be regarded as the ensemble of non-specific sites. Defining a 3D-grid, each sitebeing approximately of the volume of a ribosome, provides the statistical means for adetailed description of the degeneracy D ( j ). Similar ideas for, i.e., ligand-receptor bind-ing on a 2D-grid have motivated this approach [21]. A microstate is a realisation of thedistribution of the diffusing or free ribosomes n f into the N compartments, Fig. 1(a).The number of possible microstates for such a system is simply given by the binomialcoefficient D ( n b ) = (cid:0) Nn − n b (cid:1) .In the most general case we will consider N populations of mRNAs, each onecharacterised by a different ribosome affinity, ∆ (cid:15) ( i ) , and competing for the n f ribosomesin the reservoir. The mRNA population i contains M ( i ) transcripts, with a maximalcapacity n ( i )max corresponding to largest number of translating ribosomes that an mRNAcan fit. Hence, n ( i )max := L/(cid:96) with L being the length of the transcript in codons, and (cid:96) the footprint of the ribosomes. A typical gene has a length L = 300 codons and aribosome covers (cid:96) = 10 codons. We summarise in Table 1 the symbols used in this work. In the previous sections we revised the equilibrium model of gene expression asintroduced in [18, 19] in the context of transcription, and which has been extendedin [1, 11, 12] to multiple genes to investigate the titration of transcription factors. Herewe extend this model to consider the concurrent binding of many particles on the samesubstrate. This is a natural extension when studying the competition of genes forRNAP -in the case of transcription- and for ribosomes - in the case of translation. Asalso explained above, we will develop this model in the context of mRNA translation, n equilibrium model for ribosome competition Symbol Meaning n f Unbound, free ribosomes constituting thereservoir n ( i ) b Ribosomes bound to all mRNAs belonging tothe population in ( i )max Ribosome capacity of an individual mRNAbelonging to the population iN Number of cytoplasmic compartments (nonspecific sites) M ( i ) Number of mRNAs in population i N Number of mRNA populations n b Ribosomes bound (total): n b = (cid:80) N i =0 n ( i ) b n Total number of ribosomes: n = n f + n b ∆ (cid:15) ( i ) Energy turnover when a single ribosomebinds to an mRNA of population i Table 1.
Summary of the symbols used and their meaning. where ribosomes are considered as limiting [13] and their abundance strictly related tothe cellular physiology [14, 15, 16].In order to do that, instead of considering that each mRNA is either occupied by oneribosome ( n b = 1, actively translating state) or empty ( n b = 0, untranslating state) as itwould be in the standard thermodynamic model introduced in Section 2.1, we imaginethat a number n b ≤ n max of ribosomes can be recruited by an individual mRNA, asshown in Figure 1(a). This corresponds to assume that the RBS of the mRNA becomesimmediately available (if n b < n max ) and a new ribosome can bind again the lattice.We point out that, due to the model definitions, the recruitment capability of anmRNA decreases with its ribosome occupation as its number of available sites decreases-see Appendix A. This is somehow consistent with what other models consideringribosome interference would predict [22].In the following sections we will develop the model with multiple ribosome bindingon an individual mRNA (Section 2.3.1), then extend it to several copies of the sametranscript (Section 2.3.2) and finally extend it to the case of several mRNA typeshaving different properties. This general case depicted in Figure 1(b) will be treated inSection 2.3.3. The model revisited in Section 2.1 illustratesthe derivation of the occupation probability p for the simple case of two states: thesingle specific site being occupied or unoccupied. This case corresponds to n max = 1,and here we extend it to any value of n max in order to obtain the occupation probability p ( n b ) for a system with n max + 1 possible states. In this section we begin considering anindividual mRNA ( N = 1, M = 1) in contact with a reservoir of ribosomes. n equilibrium model for ribosome competition (a) (b) Figure 1. (a) Sketch of a possible microstate of the model, with n f free ribosomes(yellow particles) in the N cytoplasmic compartments (the white spheres) and n b = 2ribosomes on the mRNA (grey squares). With respect to the n b = 0 situation, thisconfiguration has an energy turnover n b ∆ (cid:15) . n max indicates the maximum number ofribosomes concurrently translating the mRNA (number of squares). (b) The full modeltakes into account a number N of mRNA populations (here N = 3, each one composedof M ( i ) mRNAs with the same n max ). In the case of the equilibrium thermodynamic model (Section 2.1), gene expressionlevels are proportional to p ( n b = 1), as at most one particle can bind the promoter.Instead, in our model extension with multiple particle binding we can reasonably assumethat protein synthesis levels are proportional to (cid:104) n b (cid:105) , the average number of ribosomesbound to the transcript. This is what it is usually done to interpret experimental data inmRNA translation, and a measure of translation efficiency is directly related to ribosomedensity [23].The partition functions Z ( n b ) = D ( n b ) e − β [ n b ∆ (cid:15) ] for each macrostate characterisedby n b are necessary to compute (cid:104) n b (cid:105) and are derived in Appendix C for the most generalcase. The reader can recover the results of this current section by fixing N = 1, M = 1 inthe equations of this appendix. As multiple binding can in principle be considered witha capacity n max larger than the total number of particles n , we further need to considerthat the mRNA cannot be occupied by more ribosomes than there are available forbinding. Summing up the different accessible states and by taking into account thisrestriction we obtain Z tot0 = min( n,n max ) (cid:88) n b =0 Z ( n b ) . (4)For a macrostate with partition function Z ( n b ) we find two separable parts contributingto the degeneracy D ( n b ). On the one hand, the n f free ribosomes are distributed intothe N cytoplasmic compartments. On the other hand, there is an additional distributionby the n b bound ribosomes on the mRNAs’ lattice. As the lattice can be maximallyoccupied by a number n max of particles, the possible ways of distributing n b boundribosomes is given by the binomial coefficient (cid:0) n max n b (cid:1) . The degeneracy of the macrostatethen is D ( n b ) = (cid:0) Nn − n b (cid:1)(cid:0) n max n b (cid:1) and the normalised degeneracy for the single mRNA case n equilibrium model for ribosome competition D ( n b ) := ( Nn − nb )( n max nb )( Nn ) = (cid:81) n b j =1 (cid:104) ( n − j +1)( N − n + j ) ( n max − j +1) j (cid:105) n b > n b = 0 . (5)We also recall that D ( n b ) is strictly related to the hypergeometric distribution, as theproblem studied in this section is mathematically equivalent to draw n marbles (boundribosomes) from an urn containing N + n max marbles of which N are white (cytoplas-mic compartments) and n max are black (specific ribosome binding site). We again referto Appendix C for details.The quantity (cid:104) n b (cid:105) , proportional to the translation efficiency, can hence be computedas the expected value of all n max + 1 possible states of energy with probabilities p ( n b ): (cid:104) n b (cid:105) = min( n,n max ) (cid:88) n b =0 n b p ( n b ) = min( n,n max ) (cid:88) n b =0 n b Z ( n b ) Z tot0 . (6)As an example, Figure 2 shows the behaviour of (cid:104) n b (cid:105) as a function of the total numberof ribosomes n for three exemplifying values β ∆ (cid:15) . The ∆ (cid:15) represents the strength ofthe RBS and accounts for hybridization and non-optimal spacing of ribosome subunits,unfolding of mRNA, etc. [17]. The analytical solution Eq. (6) is compared to simulations Figure 2. (a)A single mRNA type ( N = 1), one copy ( M = 1): Plot of (cid:10) n b (cid:11) as a function of n ∈ (0 , β ∆ (cid:15) ∈ {− , − , − } . Thenumber of cytoplasmic compartments is fixed to N = 10 and the maximal ribosomecapacity per mRNA is n max = 10. The solid line is obtained with the analytic solutionEq. (6), whereas the circles with error-bars are obtained by simulating the systemusing the Gillespie chemical reaction algorithm [24]. (b) occupation probability p ( n b )for β ∆ (cid:15) = − of the system with the Gillespie algorithm, whose details are explained in Appendix A.The agreement between the theory and the numerical simulation is excellent. n equilibrium model for ribosome competition M copies. For the sake of completeness we go onestep further, and extend the previous case of a single mRNA copy M = 1 to any M ≥
1. Substantially, one obtains the same Equations (4), (5) and (6) with thesubstitution n max → n max M , meaning that the demand of the transcripts for theribosomes is increased of a factor M . We define (cid:104) ¯ n b (cid:105) := (cid:104) n b (cid:105) / ( n max M ) being theoccupation (normalised) of each individual mRNA, and we also refer to (cid:104) ¯ n b (cid:105) as thetranslation efficiency of an mRNA belonging to this population.Figure 3(a) shows (cid:104) ¯ n b (cid:105) as a function of the total number of ribosomes n with a numberof M = 10 mRNA copies. In order to show the competition between different membersof the same mRNA population, we compare the outcomes with the ones obtained forthe single-copy case treated in the previous section and in Figure 2, and the analyticalsolution for M = 1 case is shown by the dashed lines. Figure 3.
Unique mRNA population N = 1, M ≥
1. Normalised transcriptoccupancy (cid:10) ¯ n b (cid:11) as a function of the total number of ribosomes n (a) for fixed M = 50,and of the total number of transcripts M (b) for fixed n = 3000. Dashed lines in panel(a) show the results for M = 1. In both panels n max = 10. So far we have considered the supply of ribosomes n as a control parameter of themodel. However, when considering multi-copy mRNAs, we can also vary the demand ofthe system, i.e. M . When the demand is low, each individual mRNA is at its maximumcapacity, and it decreases by increasing the number of mRNAs M in the population -seeFig. 3(b). N mRNA populations, M ( i ) copies. The genome codes for a large number ofdifferent types of mRNAs with unique properties as RBS strength, codon length andtranslation rate. To account for this diversity, we develop the general case of i = 1 , ..., N distinct types of mRNA populations. Where necessary, the superscript ( i ) indicates thata parameter is specific for the population i . n equilibrium model for ribosome competition p ( n ( l ) b ) of having n ( l ) b ribosomes bound to the mRNApopulation l is given by the sum of all states with the given n ( l ) b divided by Z tot : p ( n ( l ) b ) = (cid:88) n ( l ) b = const Z ( n (1) b , ..., n ( l ) b , ..., n ( N ) b ) Z tot . (7)We recall that each normalised partition function Z represents a unique configurationof energy as a function of the bound particles to the different populations (cid:80) N i =1 n ( i ) b ∆ (cid:15) ( i ) ,with Z ( { n ( i ) b } ) = D ( { n ( i ) b } ) e − β [ (cid:80) N i =1 n ( i ) b ∆ (cid:15) ( i ) ] , (8)and D ( { n ( i ) b } ) = ( Nn − nb ) (cid:81) N i =1 ( n ( i )max M ( i ) n ( i ) b )( Nn ) = (cid:81) n b j =1 (cid:104) ( n − j +1)( N − n + j ) (cid:105) (cid:81) N i =1 D ∗ ( i ) n b > n b = 0 , (9) D ∗ ( i ) = (cid:81) n ( i ) b k =1 (cid:104) n ( i ) max M ( i ) − k +1 k (cid:105) n ( i ) b > n ( i ) b = 0 , (10)where n b = (cid:80) N i =1 n ( i ) b is the total number of bound ribosomes.The degeneracy D in Eq. (9) consists of two parts: The left product correspondsto the degeneracy of the case of a single mRNA population with one copy number( N = 1 , M = 1), see Equation (5); the right part is the product over all populationsof distributing n ( i ) b particles on the n ( i ) max M ( i ) sites available for each population i , assketched in Figure 1(b). Similarly to Equation (5), this problem is similar to randomlydrawing n marbles from an urn containing (cid:80) N n ( i )max M ( i ) marbles of N different colors.The degeneracy D in Eq. (9) is thus related to the multivariate generalisation of thehypergeometric distribution. Another way to understand this is to imagine that anumber n ( l ) b of ribosomes bound to the population l can be realised by the product ofall possible distributions { n ( i ) b } i (cid:54) = l of ribosomes bound to other populations, as givenby (cid:81) N i =1 ,i (cid:54) = l D ∗ ( i ).Following the very same considerations, one finds the total partition function to be Z tot := min ( n,n (1) max M (1) ) (cid:88) n (1) b =0 · ... · min ( n − (cid:80) N− i =1 n ( i ) b ,n ( N ) max M ( N ) ) (cid:88) n ( N ) b =0 Z ( { n ( i ) b } ) . (11)Thanks to the general case developed in this section we can study the competitionbetween populations under limited resources, as their different ribosome recruitmentcapability can lead to counter-intuitive results. To facilitate the observation of suchcomplex behaviour we compare normalised values of the average protein synthesis permRNA (the translation efficiency) and per mRNA population relative to all the otherpopulations. The effects of competition between different populations can hence becomputed with the relative protein synthesis levels (cid:104) n ( l ) b (cid:105) rel = (cid:104) n b ( l ) (cid:105) (cid:80) i (cid:104) n b ( i ) (cid:105) . (12) n equilibrium model for ribosome competition N = 2 populations is shown in Fig. 4. In panel(a) we plot the efficiency (cid:104) ¯ n b (cid:105) of the two populations as a function of the total numberof ribosomes n . One can notice that, as expected, the population with the highestaffinity for ribosomes reaches its saturation faster compared to the other transcripts,and only then the second population increases their translation efficiency. This can alsobe appreciated by observing the relative total amount of ribosomes (cid:104) n ( i ) b (cid:105) rel involved intranslation of the two populations, panel (b). Figure 4.
Two competing populations. Panels (a) and (b) respectively show (cid:10) ¯ n b (cid:11) and (cid:10) n ( i ) b (cid:11) rel as a function of n , for N = 10 , n ( i ) max = 10 , M ( i ) = 50. In panels (c) and(d) we vary the amount of transcripts M (2) of the population 2. Dashed lines indicatewhen (cid:80) i n ( i ) max M ( i ) = n , i.e. when the system enters a severely limited resourcesregime. Insets in (b) and (d) show the number of free ribosomes as functions of n and M (2) respectively. Instead of varying the supply of ribosome we can also change the amount oftranscripts in each population. In panels (c) and (d) we increase M (2) and observehow the efficiency of each single transcript drops when entering a regime in which n equilibrium model for ribosome competition n < (cid:80) i n ( i ) max M ( i ) ), while the weight (cid:104) n ( i ) b (cid:105) rel of the population 2 increases at the expenses of the other population.
3. Competition regulated gene expression
Assuming that the translation rate of a transcript is proportional to the average numberof ribosomes (cid:104) n ( i ) b (cid:105) bound to the population i , the evolution of the protein concentration P ( i ) produced by the population i is given by dP ( i ) dt = k TL (cid:104) n ( i ) b (cid:105) − δP ( i ) , (13)where δ is the degradation rate of that protein, and k TL is the translation rate fixingthe timescale of protein production of each ribosome. This gives (at steady-state) aprotein concentration proportional to the total ribosomes engaged in the production ofthe protein of interest: P ( i ) = k TL δ (cid:104) n ( i ) b (cid:105) .With these prescriptions and with the theory developed in the previous sectionswe can now speculate how competition for ribosomes can in principle affect cellularphysiology and protein synthesis. We show that this model predicts that relative protein levels can be regulated by varyingthe supply or the demand of the translation machinery. Rather than absolute proteinconcentrations P ( i ) we also focus on relative expression levels P ( i )rel defined as P ( i )rel = P ( i ) (cid:80) j P ( j ) = (cid:104) n ( i ) b (cid:105) rel , (14)and we assume that the translation and degradation rate of all proteins are the same.Thus, the relative protein abundance is equivalent to the fraction of ribosomes that arebound to the population of interest i , as defined in Eq. (12).As a proof of principle, we plot the protein abundances of a system with N = 3populations. This is similar to what we have shown in Fig. 4, but now we present a morecomplex situation with more populations competing for the same reservoir of ribosomes.We show that protein synthesis levels change when changing the trade-off of ribosomalresources. When n (cid:28) (cid:80) i n ( i ) max M ( i ) the supply is smaller than the capacity of thesystem, and we are in a regime that is severely limited by the amount of ribosomes. Inthis regime all the transcripts are coupled via the pool of free ribosomes n f . Otherwise,when n (cid:29) (cid:80) i n ( i ) max M ( i ) the number of ribosomes is not limiting and asymptotically thesystem would behave as if each element is independent.In panels (a) and (b) of Fig. 5 we increase the supply of ribosomal resources n .We first plot the normalised ribosome occupation of each population, which is a proxyfor the efficiency of protein production of each single mRNA. The efficiency of each n equilibrium model for ribosome competition Figure 5.
Finite ribosomal resources affect protein synthesis of competing populations( N = 3). In panel (a) we show the transcript efficiency (cid:10) ¯ n b (cid:11) as a function of n , while inpanel (b) the resulting relative composition of proteins. M ( i ) = 20 for each populationand the energy turnover β ∆ (cid:15) is given in the legend of panel (b). Panels (c) and (d)show translation efficiency and the relative protein abundances when changing theamount of members of the population 3 (energy turnover as in the legend of panel(d)). M (1) = M (2) = 20. For all panels N = 10 , n ( i )max = 10. Panels (e) and (f)show the pie charts of the protein composition for the points indicated in (b) and (d)respectively. transcript increases non-linearly until saturating for very large n , i.e. when resourcesare no longer limiting. However, mRNAs belonging to different populations increasetheir efficiency differently as a function of the supply available. To highlight this, inpanel (b) we show the relative expression levels P ( i )rel , which is the fraction of proteinsproduced by the population i , as a function of the supply level. As the total numberof ribosomes is increased, the relative expression changes in a counter-intuitive fashion.In the supply limited regime n (cid:28) (cid:80) i n ( i ) max M ( i ) , the strongest population (the one withthe best transcript-ribosome affinity) engages most of the ribosomes, but when n isincreased its weight decreases at the advantage of the other populations. We stress that P ( i )rel can present on optimal value or, in other words, that the production of given mRNA n equilibrium model for ribosome competition M (3) ), the extra transcripts do not affect the efficiency of themRNAs belonging to the other populations -panel (c)-, but the relative weights oftheir populations are already affected -panel (d)-. In the ribosome-limited regime thecompetition becomes severe and it drastically affects the efficiency of the individualmRNAs. Under strong competition the populations behave differently according totheir recruitment capability.We also represent in panels (e) and (f) the pie charts of the protein composition ofthis three-population system for three values of n -panel (e)- and M (3) -panel (f). In the previous section we have emphasised how ribosome competition might give riseto unexplored means of gene expression regulation at the level of translation. Now wespeculate on the interplay between ribosome allocation between mRNAs of differentpopulations and cellular physiology.Following the derivation of [16], the growth rate λ of a cell is related to the biomass M and the total of actively translating ribosomes n act by d M dt = λ M = k TL n act . (15)Therefore, the growth rate is proportional to the number of ribosomes engaged intranslation. Coarse graining the entire translatome of the cell with a single populationof mRNAs, we can then identify n act with (cid:104) n (1) b (cid:105) , and thus assume that λ ∝ (cid:104) n (1) b (cid:105) .Figure 6(a) shows the relation between the total number of ribosomes and the growthrate, as predicted by our toy model. This almost linear regime, obtained with parametersthat are physiologically reasonable, is qualitatively similar to the phenomenologicalgrowth law presented in [14]. Although we notice that λ = 0 is obtained when n = 0,one could impose a number of ribosomes n min that cannot be involved in translation, n act = (cid:104) n (1) b (cid:105) − n min , as done in [16].For a fixed number of ribosomes n , and thus for a given growth rate λ , themodel predicts that when adding transcripts of an exogenous population M (2) , theproduction of their protein will be at the expenses of the endogenous population,as shown in Fig. 6(b). Transcripts that are normally engaged in the translation ofnecessary endogenous proteins compete with the exogenous population for the pool ofavailable ribosomes. The growth rate will then decrease as the demand of the competingpopulation increases, and thus as the synthesis of the exogenous population grows. Themechanism proposed might be one of the basic principles of protein burden. n equilibrium model for ribosome competition Figure 6.
Impact of ribosome competition on the cellular physiology. The relationbetween the estimated cellular growth rate and the total number of ribosomes isshown in panel (a) assuming a translatome composed of an individual population with M (1) = 2000, and energy turnover as in the legend. The inset shows the amount of freeribosomes n f as a function of n . By fixing the total amount of ribosomes to n = 3 × (green star), we imagine to start adding transcripts of an exogenous gene belongingto a second population, panel (b). The M (2) transcripts then increasingly soak upresources (i.e. the number of ribosomes n (2) b engaged in translation of the exogenouspopulation increases, triangles) that are necessary to the endogenous population, thusdecreasing the growth rate (which is proportional to n (1) b , red circles). M (1) = 2000 asin panel (a).
4. Discussion
In this work we have introduced an equilibrium model of mRNA translation toinvestigate, on theoretical grounds, the impact of ribosome competition on thetranslation machinery. We have extended the thermodynamic framework of geneexpression, so far implemented to model transcription, to the translation step.Our toy model assumes that the ribosomal pool is in equilibrium with thetranslatome, the ensemble of actively translating mRNAs, and we allow for multipleribosome binding on the same mRNA. Our coarse-grained approach neglects manybiological processes, such as ribosome biogenesis or cellular metabolism. However,even such a simple model presents a complex phenomenology, and we speculate onqualitative aspects of ribosome competition related to gene expression regulation thatare yet unexplored.In principle the theory could be exploited for any number of mRNA populations,each one having different properties. However, the system can easily become intractablenumerically because of the multiplicity terms D ( { n ( i ) b } ). Although the development ofan approximated theory is out of the scope of this work, it is always possible to exploreany possible system by means of the stochastic simulations explained in Appendix A.This modelling framework could in principle be exploited in other processes, for n equilibrium model for ribosome competition in vitro systems, where supply and demandcan be better controlled and decoupled from the cellular physiology.
5. Acknowledgements
LC and PSR would like to thank the Erasmus exchange program, thanks to which PSRcould spend time at the University of Montpellier working on this project, and Prof.Joerg Fitter who was PSR’s contact at the origin institute. TJR acknowledges FondecytIniciacion 11161046 for funding.
References [1] Robert C. Brewster, Franz M. Weinert, Hernan G. Garcia, Dan Song, Mattias Rydenfelt, and RobPhillips. The Transcription Factor Titration Effect Dictates Level of Gene Expression. Cell,156(6):1312–1323, 3 2014.[2] Hidde de Jong, Johannes Geiselmann, and Delphine Ropers. Resource Reallocation in Bacteria byReengineering the Gene Expression Machinery. Trends in Microbiology, 25(6):480–493, 6 2017.[3] Francesca Ceroni, Alice Boo, Simone Furini, Thomas E. Gorochowski, Olivier Borkowski, Yaseen N.Ladak, Ali R. Awan, Charlie Gilbert, Guy Bart Stan, and Tom Ellis. Burden-driven feedbackcontrol of gene expression. Nature Methods, 2018. n equilibrium model for ribosome competition [4] Alexander P.S. Darlington, Juhyun Kim, Jos´e I. Jim´enez, and Declan G. Bates. Dynamic allocationof orthogonal ribosomes facilitates uncoupling of co-expressed genes. Nature Communications,2018.[5] Renana Sabi and Tamir Tuller. 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Self-consistent theory of transcriptional control in complex regulatory architectures. PLOS ONE,12(7):e0179235, 7 2017.[11] Mattias Rydenfelt, Robert Sidney Cox, Hernan Garcia, and Rob Phillips. Statistical mechanicalmodel of coupled transcription from multiple promoters due to transcription factor titration.Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 2014.[12] Franz M. Weinert, Robert C. Brewster, Mattias Rydenfelt, Rob Phillips, and Willem K. Kegel.Scaling of gene expression with transcription-factor fugacity. Physical Review Letters, 2014.[13] Premal Shah, Yang Ding, Malwina Niemczyk, Grzegorz Kudla, and Joshua B. Plotkin. Rate-limiting steps in yeast protein translation. Cell, 153(7):1589–1601, 2013.[14] Matthew Scott, Eduard M Mateescu, Zhongge Zhang, and Terence Hwa. Interdependence of CellGrowth Origins and Consequences. Science, 330:1099–1102, 2010.[15] Matthew Scott and Terence Hwa. Bacterial growth laws and their applications. 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Nonspecific transcription factor binding can reduce noisein the expression of downstream proteins. Physical Biology, 12(5):055002, 8 2015.[27] Ron Milo and Rob Phillips. Cell Biology by the Numbers. Garland Science, 2016.[28] Adriana Verschoor, Jonathan R. Warner, Suman Srivastava, Robert A. Grassucci, and JoachimFrank. Three-dimensional structure of the yeast ribosome. Nucleic Acids Research, 1998. Appendix A. Supporting the analytical solutions using simulations basedon the Gillespie algorithm
We implement a simulation scheme based on the ‘direct method’ version of Gillespie al-gorithm [24, 11] to support the analytical results obtained in Section 2.3 and computedby numerically solving the equations throughout the text. By implementing this chemi-cal reaction algorithm that connects reacting species (ribosomes and mRNAs) with thecorresponding chemical reactions (un/binding of ribosomes and mRNAs), we obtain astochastic simulation in excellent agreement with the rather complicated and detailedanalytical solutions. The Gillespie algorithm draws the time τ of the next possible reac-tion from an exponential distribution e − Aτ with mean equal to the inverse of the sum ofall the reaction rates of possible events A = (cid:80) j a j , with a j being the rate of the event j - also named propensity. The subsequent reaction r is then again randomly drawn fromthe uniform distribution with probability density function 1 /A .In order to derive the necessary kinetic rates we first focus on the case N = M = n max = 1. The reaction between the unbound (with n free ribosomes) and bound (with n − n ; n b = 0 k on (cid:10) k off n − n b = 1 (A.1)where k on and k off are reaction rate constants. Hereby, k on is the rate of number ofassociations per free ribosomes, per mRNA, and k off is the rate of disassociation of thebound particle, per free compartments. We find these constants by solving the chemicalmaster equation of the probability of occupation p ( n b = 1): ddt p ( n b = 1) = k on p ( n b = 0) n − k off p ( n b = 1)( N − n + 1) . (A.2)At the steady state, with the occupation probability p according to the case describedin Section 2.3.1 we obtain the relation k on = k off ( N − n + 1) n p ( n b = 1) p ( n b = 0) = k off e − β ∆ (cid:15) . (A.3)The exponential weight accounts for the RBS strength, meaning that differentpopulations i of mRNAs react on different time scales due to specific k ( i ) on (∆ (cid:15) ( i ) ).After having computed the kinetic constants k ( i ) on and k ( i ) off for each population i , and n equilibrium model for ribosome competition α of the population i can be written as n f ; n ( i,α ) b k ( i ) on (cid:10) k ( i ) off n f − n ( i,α ) b + 1 , (A.4)with n ( i,α ) b < n ( i ) max . In the previous equation we have introduced the notationwith the index ( i, α ) to highlight the binding/unbinding event for each mRNA α ,and we emphasise that the rate constants are the same for all the mRNAs in thepopulation i . Thus the individual propensities are a ( i,α ) on = n f k ( i ) on ( n ( i ) max − n ( i,α ) b ) and a ( i,α ) off = k ( i ) off n ( i,α ) b ( N − n f ). For the sake of simplicity one could also write thealgorithm in terms of the population i , without distinguishing the individual mRNAs.In this case the propensities for each population are a ( i ) on = n f k ( i ) on ( n ( i ) max M ( i ) − n ( i ) b ) and a ( i ) off = k ( i ) off n ( i ) b ( N − n f ).The algorithm follows the time evolution of the chemical species n ( i,α ) b . After atransient time, the abundances of species tend to equilibrium. We then calculate thetime average over several intervals of the same length and compute the average valueand its standard deviation. We choose 10 iteration steps to overcome the transientregion, and calculate the average value with 10 intervals of 10 iterations. Appendix B. Determination of N For comparison with a physiological relevant system, we suggest an approximate valueof the number of boxes N for the bacterial E.coli . In Section 2.1 we mention the cells’cytoplasm for distributing the free ribosomes but did not go into a more detailed de-scription, which we provide here.In order to compute the number of available boxes we consider that the cytoplasmicvolume V cyto contributes to the total number of boxes. Reasonably assuming ribosomesas spheres, the effective stacking of these (see Figure 1 (a)) leads to about 30% of spacenot accessible for distribution, by which we correct to 0 . V cyto of available space for thespheres. We further approximate that V cyto ≈ V cell , where V cell is the volume of the cell,as the width of the membrane is negligible in comparison with the ‘radius’ of the cell.Hence N ≈ . V cell /V ribo ), where V ribo is the volume of a ribosome.The cell volume V cell ≈ . µ m corresponds to n ≈ · ribosomes [27]. With theribosome volume V ribo ≈ . · − µ m [28], we then find N ≈ . · , whereby about20% is already occupied by the n ribosomes. n equilibrium model for ribosome competition Appendix C. Derivation of the mathematical results as given in the generalcase
In Section 2.1 we introduced the partial partition function Z . Here we derive thenormalized partial partition function Z for the general case with N populations eachone composed of M ( i ) transcripts. Z = Z ( { n ( i ) b } ) is a function of the bound particles n ( i ) b to all populations i and determined by the sum of all microstates (the degeneracy D ) with the same exponential weight (according to the systems energy state E ): Z ( { n ( i ) b } ) = D ( { n ( i ) b } ) e − βE ( { n ( i ) b } ) . (C.1)One can find the energy of a state with { n ( i ) b } by summing up the energetic contributionsof bound and unbound particles, with (cid:15) b and (cid:15) f respectively, to be E ( { n ( i ) b } ) =[ n f (cid:15) f + (cid:80) N i =1 n ( i ) b (cid:15) ( i ) b ]. Using n f = n − n b = n − (cid:80) i n ( i ) b one finds E ( { n ( i ) b } ) =[ n(cid:15) f + (cid:80) N i =1 n ( i ) b ( (cid:15) ( i ) b − (cid:15) f )] = [ n(cid:15) f + (cid:80) N i =1 n ( i ) b ∆ (cid:15) ( i ) ], with the change in energy ∆ (cid:15) ( i ) given by the chemical reaction between a ribosome and an mRNA of population i .The contributions to the non-normalized degeneracy D ( { n ( i ) b } ) are, as described inSection 2.1, on the one hand the degeneracy of distribution n f = n − n b of free particlesinto the cytoplasm with N boxes, and on the other hand the product of the degeneracyof n ( i ) b bound ribosomes on the n ( i ) max M ( i ) sites of population i . This gives D ( { n ( i ) b } ) = (cid:18) Nn − n b (cid:19) N (cid:89) i =1 (cid:18) n ( i ) max M ( i ) n ( i ) b (cid:19) . (C.2)As mentioned above, the degeneracy of the populations are decoupled (leading to theproduct over the populations), as a change of particle between populations leads to achange of E , representing different macrostates.Plugging D and E in Equation (C.1) we obtain the partial partition function of thegeneral case of i = 1 , ..., N different populations with M ( i ) copies Z ( { n ( i ) b } ) = (cid:18) Nn − n b (cid:19) N (cid:89) i =1 (cid:18) n ( i ) max M ( i ) n ( i ) b (cid:19) e − β [ n(cid:15) f + (cid:80) N i =1 n ( i ) b ∆ (cid:15) ( i ) ] . (C.3)To normalise Equation (C.3) it seems reasonable to choose as reference state n b = 0 forwhich { n ( i ) b } = { } . With the corresponding Z ( { } ) = (cid:18) Nn (cid:19) e − β [ n(cid:15) f ] . (C.4)one obtains Z ( { n ( i ) b } ) = Z ( { n ( i ) b } ) Z ( { } ) (C.5)= (cid:0) Nn − n b (cid:1)(cid:0) Nn (cid:1) N (cid:89) i =1 (cid:18) n ( i ) max M ( i ) n ( i ) b (cid:19) e − β [ (cid:80) N i =1 n ( i ) b ∆ (cid:15) ( i ) ] (C.6)= D ( { n ( i ) b } ) e − β [ (cid:80) N i =1 n ( i ) b ∆ (cid:15) ( i ) ] , (C.7) n equilibrium model for ribosome competition D is the degeneracy refered to state n b = 0. The quantity D has further beenreduced and expressed as given in Equations (9) and (10) of Section 2.3.3 to make itmore accessible for computational purposes. For doing so, one applies the followingrelations(1) b !( b − c )! = b · ( b − · ... · ( b − c + 1) = (cid:40) (cid:81) cj =1 ( b − j + 1) c > c = 0(2) ( a − b )!( a − ( b − c ))! = 1( a − b + 1) · ... · ( a − b + c ) = (cid:40) (cid:81) cj =1 1 a − b + j c > c = 0(3) 1 c ! = 11 · · ... · c = (cid:40) (cid:81) cj =1 1 j c > c = 0 . Appendix D. Cell-to-cell variability of ribosomal resources
Although in this work we focused on fixed values of resources (constant number ofribosomes), in this Appendix we perform a simulation by considering a population ofisogenic cells whose ribosome content is Gaussian distributed (and mRNA is constant).Figure D1 shows the outcome of this simulation, which corresponds to the situationshown in Figure 5 with normally distributed total amount of ribosomes n (standarddeviation 5%). Appendix E. Alternative mRNA degeneracy
In the main text we have coarse-grained each mRNA population and characterised itas a 2D-lattice -see Fig 1 (b)- with n ( i ) max M ( i ) accessible sites. With this choice wecannot follow each transcript individually, and the macrostate of a population i with n ( i ) b ribosomes bound has a degeneracy (cid:0) n ( i ) max M ( i ) n ( i ) b (cid:1) .However, other representations of ribosome recruitment and of the mRNApopulation are possible. For instance, we could refine the treatment of the mRNApopulations by considering each transcript independently, which is equivalent toalternatively represent the degeneracy terms D ∗ .In order to extend the model and explicitly consider each transcript in a population,we recall a well known problem in combinatorics: “We consider two dice each havingsix different numbers of eyes. What is the number of different ways to roll the two dice,given that we are interested in the sum of the eyes? Note that the order of rolling thedice is not of concern for our considerations.” This problem can be translated into thelanguage of our problem: The sum of the eyes is the quantity n ( i ) b , the number of diceis M ( i ) and the number of different numbers of eyes is n ( i ) max + 1. In the formulation ofthe problem used in the text we substantially defined the problem with a single diceper population. Instead, the solution of this combinatorial problem has been found by n equilibrium model for ribosome competition Figure D1.
Equivalent of Figure 5 but with normally distributed n , standarddeviation 5%. Each point is the result of 100 simulations with n drawn from thisdistribution. Abraham de Moivre a few centuries ago, and we obtain: D ∗ ( i ) = (cid:98) n ( i ) b / ( n ( i ) max +1) (cid:99) (cid:88) k =0 ( − k (cid:18) M ( i ) k (cid:19) (cid:18) n ( i ) b + M ( i ) − k ( n ( i ) max + 1) − M ( i ) − (cid:19) . (E.1)We have tested this approach in a small system composed of N = 3 populationseach one composed of M ( i ) = 5 mRNAs as shown in Figure E1. In panel (a) we show therelative ribosomes per mRNA population (cid:104) n b (cid:105) rel as computed with the model definitionsused in the main text. Instead, in panel (b) we compute the same quantity with thedegeneracy introduced in this appendix. When the system is in a regime not limitedby the resources (large n ), the two models converge to the same result. However, thereare evident differences when ribosomes are limiting. If further extensions are out of thescope of this paper, here we show that small changes in the model can largely impact n equilibrium model for ribosome competition Figure E1.
Relative ribosomes per population with the model definitions used in themain text -panel (a)- and with the prescriptions given in this appendix -panel (b)-. N = 10 , n maxmax