Modelling the effect of ribosome mobility on the rate of protein synthesis
Olivier Dauloudet, Izaak Neri, Jean-Charles Walter, Jérôme Dorignac, Frédéric Geniet, Andrea Parmeggiani
MModelling the effect of ribosome mobility on the rate of protein synthesis
Olivier Dauloudet , , † , Izaak Neri , † , ∗ , Jean-Charles Walter , † , ∗ ,J´erˆome Dorignac , Fr´ed´eric Geniet , Andrea Parmeggiani , ∗ Laboratoire Charles Coulomb (L2C), Montpellier University, CNRS, Montpellier, France Laboratory of Parasite Host Interactions (LPHI),Montpellier University, CNRS, Montpellier, France Department of Mathematics, Kings College London, Strand, London, WC2R 2LS, UK and † These authors contributed equally. (Dated: October 1, 2020)Translation is one of the main steps in the synthesis of proteins. It consists of ribosomes thattranslate sequences of nucleotides encoded on mRNA into polypeptide sequences of amino acids. Ri-bosomes bound to mRNA move unidirectionally, while unbound ribosomes diffuse in the cytoplasm.It has been hypothesized that finite diffusion of ribosomes plays an important role in ribosome re-cycling and that mRNA circularization enhances the efficiency of translation, see e.g. Ref. [1]. Inorder to estimate the effect of cytoplasmic diffusion on the rate of translation, we consider a TotallyAsymmetric Simple Exclusion Process (TASEP) coupled to a finite diffusive reservoir, which wecall the Ribosome Transport model with Diffusion (RTD). In this model, we derive an analyticalexpression for the rate of protein synthesis as a function of the diffusion constant of ribosomes, whichis corroborated with results from continuous-time Monte Carlo simulations. Using a wide range ofbiological relevant parameters, we conclude that diffusion in biological cells is fast enough so thatit does not play a role in controlling the rate of translation initiation.
I. INTRODUCTION
Cells synthesize proteins by first transcribing the hereditary information encoded in genes into functional mRNA,and subsequently by translating the mRNA nucleotide sequence into polypeptide sequences [1]. The translation ofmRNA into a polypeptide sequence can be divided into three stages, namely, the initiation, elongation and terminationstages [1]. During initiation, a ribosomal complex (consisting of two ribosomal subunits, initiation factors, and tRNA)is assembled at the 5’ end of a mRNA chain. After initiation, the ribosomal complex moves (or elongates) from the 5’end towards the 3’ end of the mRNA while forming a polypeptide chain. In the final termination stage, the ribosomecomplex releases the polypeptide chain, unbinds from the mRNA and dissasembles.Translation is mainly controlled at the initiation step, as it is the rate limiting step in translation [2–5]. Initiationis a complex process involving several molecular actors, and it is therefore difficult to understand all the molecularmechanisms that are relevant for translation control. Nevertheless, coarse-grained mathematical modelling can uncoverwhich physical mechanisms play a role in translation control.It has been argued that the recycling of ribosomes through Brownian diffusion in the cytosol plays an importantrole in the control or regulation of translation [1, 6–8]. When a ribosome unbinds from the mRNA after termination,it can either rebind to the same mRNA or bind to another mRNA. If the diffusion of ribosomes is slow enough, thencircularization of the mRNA could enhance the rate of ribosome recycling through cytosolic diffusion [1, 6, 9, 10].On the other hand, this effect would be negligible if diffusion of ribosomes is fast enough. In this paper we usephysical modelling to determine whether recycling of ribosomes through diffusion can play a role in controlling mRNAtranslation.In order to study how ribosome mobility affects the mRNA initiation rate and thus the protein production, wepresent a minimalistic physical model that describes both the translation of mRNA by ribosomes and the diffusion ofribosomes in the cytoplasm. We call this model the Ribosome Transport model with Diffusion (RTD). From a physicalviewpoint, the RTD consists of particles (the ribosomes) that diffuse in a box and can bind to a one-dimensionalsubstrate (mRNA). Particles bound to the substrate move unidirectionally and cannot overtake. The RTD consiststhus in a Totally Asymmetric Simple Exclusion Process (TASEP) [11] in contact with a diffusive reservoir. If diffusionis fast enough, then we recover the standard TASEP model, which describes in detail the elongation stage of mRNAtranslation [12–15]. On the other hand, when diffusion is slow, then a concentration gradient is formed in the reservoirand there will be a tight coupling between active transport on the filament and diffusion in the reservoir. In thisregime, the RTD describes the interplay of active and passive transport in cellular media, leading to the formation ∗ Electronic address: [email protected]; Electronic address: [email protected]; Electronic address: [email protected] a r X i v : . [ q - b i o . S C ] S e p β r r r α r β pD ˜ α FIG. 1:
Graphical illustration of the Ribosome Transport with Diffusion model (RTD).
The mRNA is represented with a dashedline, ribosomes processing along the mRNA at a rate p are represented by dark blue discs, and ribosomes diffusing freely at adiffusion coefficient D are represented by light blue discs. Grey discs of radius r centered at the end poinst of the mRNA arethe reaction volumes: if a diffusing ribosome is located in the reaction volume at the mRNA end-point centred around position r α , then it attaches at a rate ˜ α to the mRNA. On the other hand, if a ribosome is at the last site of the mRNA, then it detachesat a rate β and is released inside the reaction volume centred around r β . of a gradient of molecular species. Phenomena of active transport coupled to a diffusive reservoir have been studiedbefore in the literature, see for example Refs. [6–8, 16–25, 35, 61]. In these studies, much focus has been put onnonequilibrium phase transitions [11, 21, 26, 27].In the present paper, we use mean-field theory to derive an analytical expression for the protein synthesis in theRTD model, which is corroborated with numerical results obtained from continuous-time Monte Carlo simulations.Subsequently, we use the analytical expression for the protein synthesis rate to discuss the biological relevance ofBrownian diffusion in ribosomal recycling. By considering a broad range of biological parameters, we come to theconclusion that under physiological conditions finite diffusion of ribosomes is not important in the control of mRNAtranslation. Thus, circularisation should not occur in order to prevent the limiting effect of Brownian diffusion ofribosomes in the cytoplasm on initiation of translation [1, 6, 9, 10]. In addition, we discuss how the spatial dimensionsof the reservoir and geometry impact the protein synthesis rate and we find qualitative difference in the dependence ofthe protein synthesis rate on the length of the mRNA between two and three dimensions. Both cases are biologicallyrelevant: the three-dimensional case applies to cytoplasmic translation, whereas the two-dimensional case applies toendoplasmic reticulum translation.The paper is organized as follows. In Sec. II, we define the RTD model. In Sec. III, we present a mean-field theoryfor the RTD model and derive analytical expressions for the protein synthesis rate as a function of the diffusioncoefficient of ribosomes. In Sec. IV, we compare theory with simulations results using a continuous-time algorithm.In Sec. V, we discuss the biological relevance of the model. We conclude the paper with a discussion in Sec. VI, andin Appendix A we present analytical results for the concentration profile of ribosomes in the cytoplasm. II. MODEL DEFINITION: RIBOSOME TRANSPORT WITH DIFFUSION
We introduce here the RTD, a minimalistic model that allows us to study how diffusion determines the rate ofprotein synthesis. The RTD consists of ribosomes that diffuse in a medium embedded in two or three dimensions andcan bind to a one-dimensional substrate, say a mRNA filament. Bound ribosomes then move unidirectionally alongthe filament by converting the intracellular chemical energy from the hydrolysis of guanine triphosphate (GTP) intomechanical motion, which is modelled by a Totally Asymmetric Simple Exclusion Process (TASEP). In Fig. 1, wepresent an illustration of the model and its parameters.We consider a filament immersed in a medium containing ribosomes at a concentration c ∞ . The filament is amonopolymer consisting of (cid:96) monomers of length a . The first and last monomers of the filament are located atpositions r α and r β , respectively. For simplicity, we consider that r α and r β are fixed in time.The dynamics of unbound molecular motors is modelled as a Brownian motion with diffusion coefficient D .The dynamics of bound molecular motors is a unidirectional, hopping process with excluded volume interactions,which we model with a TASEP on a one-dimensional lattice of length L = (cid:96)a [12, 13, 26, 27]. The TASEP model isa Markov jump process with the following rates: the hopping (or elongation) rate p at which particles make a step oflength a , the exit rate β at which particles detach from the filament end-point, and the entry rate α ( t ) = ˜ α N r ( t ) , (1)where ˜ α is the rate at which ribosomes contained in the reaction volume bind to the filament and N r ( t ) is the numberof ribosomes present in the reaction volume at time t . The reaction volume is considered to be a sphere (in threedimensions) or a disc (in two dimensions) of radius r centered around the first monomer of the filament located at r α . The reaction volume radius is of the same order of magnitude as the size of a ribosome. When ribosomes detachfrom the filament they appear at a random location in a sphere (in three dimensions) or disc (in two dimensions) ofradius r centered around r β . Because of excluded volume interactions, each monomer can be bound to at most oneribosome. Therefore, ribosomes cannot hop forward if the subsequent monomer is already occupied by a ribosomeand ribosomes cannot bind to the first monomer when it is already occupied, as illustrated in Fig. 1. III. MEAN-FIELD THEORY FOR COUPLING OF DIFFUSION WITH ACTIVE TRANSPORT
We present a mean field theory for the RTD model that couples diffusion with active transport. First, in Sec. III A,we discuss how the protein synthesis rate is related to the stationary current of the TASEP model. Second, inSec. III B, we derive an analytical expression for the protein synthesis rate that is independent of the geometricalproperties of the medium or reservoir in which the one-dimensional substrate is immersed. Lastly, in Sec. III C, wediscuss the impact of the geometry of the surrounding reservoir on the protein synthesis rate.
A. Protein synthesis rate is given by the stationary current on the filament
The quantity of interest from a biological point of view is the protein synthesis rate J , which corresponds with thestationary current of particles on the filament [12, 13].The stationary current of the RTD model in the limit of infinitely large D is equal to the stationary current J ofthe TASEP model. In the limit of large (cid:96) , it holds that [11, 26, 36] J = α (cid:16) − αp (cid:17) , α < β and α < p/ , (LD) ,β (cid:16) − βp (cid:17) , β < α and β < p/ , (HD) , p , α ≥ p/ β ≥ p/ , (MC) . (2)The three branches in Eq. (2) correspond with three nonequilibrium phases: a Low-Density phase (LD) at small entryrates α < β and α < p/
2, a High-Density phase (HD) at small exit rates β < α and β < p/
2, and a Maximal Currentphase (MC) when both α ≥ p/ β ≥ p/
2. In the LD phase, the ribosome attachment process is rate limiting andthe current is a function of α ; in the HD phase, the ribosome detachment process is rate limiting and the current isa function of β ; and in the MC phase, the filament hopping process is rate limiting and the current is independentof both α and β . Experimental data in yeast cells [33] and in neurons of mammals [34] show that the rate limitingprocess for translation is the initiation of ribosomes.In the RTD model at finite values of D , the entry rate α ( t ) on the filament is not a constant but a fluctuatingquantity, see Eq. (1). In the stationary state, the average current J is well approximated by the expression (2) withthe entry rate α replaced by its average value (cid:104) α ( t ) (cid:105) = ˜ α (cid:104) N r ( t ) (cid:105) , (3)where (cid:104)·(cid:105) denotes the average over many realizations of the stationary process. Since in the stationary state theaverage number (cid:104) N r ( t ) (cid:105) of ribosomes in the reaction volume is independent of time, we set (cid:104) α ( t ) (cid:105) = (cid:104) α (cid:105) . (4)Replacing in Eq. (2) α by (cid:104) α (cid:105) , which is a mean-field assumption, we obtain for the stationary current of the RTDmodel the expression J = (cid:104) α (cid:105) (cid:16) − (cid:104) α (cid:105) p (cid:17) , (cid:104) α (cid:105) < β and (cid:104) α (cid:105) < p/ , (LD) ,β (cid:16) − βp (cid:17) , β < (cid:104) α (cid:105) and β < p/ , (HD) , p , (cid:104) α (cid:105) ≥ p/ β ≥ p , (MC) . (5)From Eq. (5) we observe that if the filament is in the HD or MC phase, then the protein synthesis rate is independentof the diffusion process in the reservoir. However, in the LD phase when the initiation step is rate limiting, which isbiologically relevant case, the current J depends on the concentration of unbound ribosomes through (cid:104) α (cid:105) , and hencein this regime we are required to include diffusion into our theoretical analysis. Often it will be insightful to considerthe limiting case where particle excluded volume on the filament is irrelevant for which the simpler formula J = (cid:104) α (cid:105) (6)holds. Note that this condition is fulfilled for low density of ribosomes on the filament. B. Protein synthesis rate: universal expression
From the point of view of the reservoir of diffusing ribosomes the filament serves both as a sink and a source ofribosomes.If the initiation and termination sites overlap, as will be approximately the case for circular mRNA, then theconcentration of ribosomes in the reservoir will be homogeneous since source and sink exactly compensate for eachother, and therefore in this case (cid:104) α (cid:105) = α ∞ = ˜ αc ∞ V , (7)where V is the reaction volume of radius r , which for two dimensions and three dimensions is given by V = πr and V = 4 πr /
3, respectively.On the other hand, if the termination site is distant from the initiation site, then (cid:104) α (cid:105) will have a reduced value, withrespect to Eq. (7) due to the depletion of ribosomes in the reaction volume at the initiation site. Indeed, the currenton the filament carries away ribosomes from the reaction volume, which in the stationary state will be compensated bythe diffusive current in the reservoir. As we will show in the next section, the depletion effects due to finite diffusionare captured by the formula (cid:104) α (cid:105) = α ∞ (cid:18) − Jµ d D eff α ∞ (cid:19) , (8)where µ d is a constant that depends on the geometry of the problem and where D eff = D ˜ αr (9)is an effective diffusion coefficient. The dimensionless quantity D eff quantifies the competition between injection ofribosomes on the filament and the diffusion of ribosomes into the reaction volume. Equation (8) follows from solvingthe diffusion equation for ribosomes in the reservoir, as we shall describe in detail in the next section. Eq. (8) statesthat the rate (cid:104) α (cid:105) is the sum of the entry rate α ∞ for a homogeneous reservoir minus a correction term that capturesthe effect of finite diffusion on the entry rate. The correction term is negative since the filament depletes particlesin the reaction volume at the initiation site. Moreover, Eq. (8) states that the correction term is proportional to thecurrent J on the filament, inversely proportional to the effective diffusion constant D eff , and it is also proportionalto the dimensionless, nonuniversal constant µ d that depends, as we shall see in the next section, on the geometricalproperties of the system, namely, the end-to-end distance | r β − r α | , the location of the filament in the reservoir, thedimensionality of the system, and the boundary conditions of the reservoir of diffusing ribosomes. Here, we wouldlike to focus on the physical consequences of the Eq.(9).To obtain the protein synthesis rate J , we combine Eqs. (5) and (8). In the LD phase, we obtain a second-orderalgebraic equation whose solution (cid:104) α (cid:105) ∈ [0 , p/
2] is given by (cid:104) α (cid:105) = p D eff + µ d µ d (cid:16) − (cid:112) − ζ (cid:17) , (10)where the adimensional parameter ζ = α ∞ D eff µ d p ( D eff + µ d ) (11)quantifies the effect of exclusion on (cid:104) α (cid:105) . The argument of the square root in (10) is always positive when the filamentis in the LD phase because in the LD phase (cid:104) α (cid:105) = p D eff + µ d µ d < p/
2, which implies ζ < /
4. Note that if the diffusion d = 0 d = D eff d = 5 D eff d = 0 d = D eff d = 5 D eff LD HDMC β/p α ∞ /p α ∞ /pJ/p (a) (b) FIG. 2: Panel (a): phase diagram for the RTD model for three values of the parameter µ d /D eff . Panel (b): protein synthesisrate J/p in the RTD model as a function of the ratio α ∞ /p for a large exit rate β > p/ coefficient D eff is small enough, then ζ (cid:28) (cid:104) α (cid:105) inside the expression forthe current, given by Eq. (5), we obtain the following expression for the protein synthesis rate, J = (cid:104) α (cid:105) (1 − (cid:104) α (cid:105) /p ) , α ∞ < β (cid:104) µ d D eff (1 − β/p ) (cid:105) and α ∞ < p/ (cid:16) µ d D eff (cid:17) , (LD) ,β (cid:16) − βp (cid:17) , α ∞ > β (cid:104) µ d D eff (1 − β/p ) (cid:105) and β < p/ , (HD) , p , α ∞ ≥ p/ (cid:16) µ d D eff (cid:17) and β ≥ p/ , (MC) , (12)where (cid:104) α (cid:105) is given by (10). For small values of ζ , we obtain the simpler expression J = α ∞ D eff D eff + µ d , (13)which also follows from Eq. (6). Equation (12) implies that the current J admits a universal expression that onlydepends on four parameters: the entry rate α ∞ for a homogeneous reservoir, the elongation rate p , the exit rate β ,and the parameter µ d /D eff that quantifies the effect of finite diffusion on the current J . From Eqs.(12) and (13) italso follows that the effect of finite mobility of ribosomes on the protein synthesis rate J is significant when µ d (cid:29) D eff .On the other hand, when µ d (cid:28) D eff , then the finite mobility of ribosomes will be irrelevant for J .In Fig. 2(a), we present the phase diagram for the RTD model for three values of µ d /D eff , namely, the case withan infinite diffusion rate, µ d /D eff = 0, and two cases with finite diffusion rates, µ d = D eff and µ d = 5 D eff . For µ d /D eff = 0, we recover the phase diagram of TASEP [11, 26, 36], while for finite values of µ d we observe an increaseof the LD phase and a corresponding decrease of the MC and HD phases. This is because finite diffusion depletesparticles in the reaction volume surrounding the initiation site of the filament, and hence reduces the current on thefilament for a given α ∞ . This is shown in Fig. 2(b), where we plot the current as a function of α ∞ /p for fixed a valueof µ d /D eff and β/p ≥ /
2. If µ d (cid:28) D eff , then the reservoir is homogeneous and we obtain the standard TASEP result[11, 26, 36] J = (cid:26) α ∞ (1 − α ∞ /p ) , α ∞ < p/ ,p/ , α ∞ > p/ . (14)In the opposing limiting case when µ d (cid:29) D eff the reservoir is strongly inhomogeneous and we obtain that J = (cid:26) D eff α ∞ µ d , α ∞ < pµ d / ,p/ , α ∞ > pµ d / . (15)In this limit the environment is viscous and therefore the effects of excluded volume become negligible.Note that the results of Fig.2 do not consider the effects of finite resources. Therefore, it is implicitely assumedthat the number of ribosomes is very large compared to the average number of ribosomes on the mRNA. In the caseof finite resources, the phase diagram displays an extended shock phase, as shown in Refs. [35, 62].So far, much of the interesting physics has been hidden in the dimensionless constant µ d that depends on thegeometry of the problem. In the next subsection we will explicitly solve the diffusion equation coupled to directedtransport on the filament to obtain explicit expressions for µ d . Z + - ---- --- ----- -- --- - ---- + +++ ++++ ++++ +++ + ++ + +++++-- L x L y FIG. 3:
Illustration of the method of images : Diffusion of ribosomes in a confined rectangular box is equivalent to diffusion ofribosomes in a two-dimensional Euclidean space that contains an infinite number of images of the original source (denoted byred) and sink (denoted by green) located in the rectangular box (located in the center and coloured in blue).
C. Influence of geometry on the protein synthesis rate
In order to obtain an expression for µ d , and thus complete the theoretical treatment for ribosomes with finitemobility, we solve the diffusion equation in the reservoir coupled with active transport on the filament. We considerthe case where | r β − r α | > r so that the reaction volumes at the source and the sink do not overlap.The stationary concentration of unbound ribosomes is described by the diffusion equation D ∆ c ( r ) = Π( r ) , (16)where c ( r ) is the concentration of ribosomes at the spatial coordinate r ∈ R d , ∆ is the Laplacian with respect theradius r , and Π( r ) = − J V | r − r α | ≤ r, J V | r − r β | ≤ r, | r − r α | > r and | r − r β | > r, (17)where we have used that | r β − r α | > r .The diffusion equation admits the solution c ( r ) = (cid:90) R d d d r (cid:48) Π( r (cid:48) ) G d ( r , r (cid:48) ) , (18)where G ( r , r (cid:48) ) is the Green function that solves D ∆ G d ( r , r (cid:48) ) = δ ( r − r (cid:48) ) . (19)The entry rate (cid:104) α (cid:105) is related to the stationary concentration in the reaction volume through (cid:104) α (cid:105) = ˜ α (cid:90) | r − r α |≤ r c ( r )d r . (20)The explicit form of the Green’s function and thus (cid:104) α (cid:105) depend on the geometry of the reservoir. We provide belowa couple of examples.
1. RTD in two-dimensional infinite box ( R ) In two dimensions, the Green function takes the form [56, 57] G ( r , r (cid:48) ) = − π ln | r − r (cid:48) | . (21)Substituting the Green function in Eq. (18), we obtain an explicit expression for c ( r ), see Appendix A. Subsequently,substituting the explicit solution for c ( r ) in Eq. (20) we obtain the formula Eq. (8) with µ = log d αβ + 12 , (22)where d αβ = | r β − r α | r (23)is the effective distance between the initiation site and the termination site on the filament. Substitution of µ d intoEqs. (10-12) provides us with an explicit expression for the current J as a function of d αβ .In Fig. 4, we plot the current J as a function of the separation d αβ between the two end-points of the mRNA fortwo values of the effective diffusion constant D eff . Although the part for d αβ < J = α ∞ (1 − α ∞ /p ) for d αβ = 0, which in Fig. 4 corresponds to J = 0 . p . We observe that the currentdecreases monotonically as function of d αβ and approaches zero for d αβ large enough. The decay towards zero islogarithmically slow after a fast initial decay in the regime d αβ <
2. RTD in three-dimensional infinite box ( R ) In three dimensions the Green function is given by G ( r , r (cid:48) ) = 14 π | r − r (cid:48) | . (24)Using this expression for the Green function in Eq. (18), we obtain an explicit expression for c ( r ), see Appendix A,which we substitute in Eq. (20) to obtain formula Eq. (8) with now µ = 25 − d αβ . (25)Comparing Eqs. (22) and (25), we see that there is a difference between two and three dimensions: in three dimensions µ converges to a finite value for d αβ → ∞ whereas in two dimensions µ diverges for d αβ → ∞ . This implies that intwo dimensions J converges to zero for large distances d αβ between the end-points of the filament, while it convergesto a finite nonzero value in three dimensions.The distinction between the dependency of the current J in two and three dimensions is illustrated in Fig. 4. Inthree dimensions, the current saturates fast to its asymptotic value after an initial quick decay for values d αβ < J depends on the diffusion constant D eff and decreases to zero for D eff →
0. Hence, in threedimensions, the mRNA will carry a finite current, even when d αβ → ∞ , and this asymptotic current will depend onthe diffusion constant.In Fig. 5, we plot the asymptotic current J as a function of the effective diffusion constant D eff . We observe fromFig. 5 that at finite D eff the protein synthesis rate in d = 2 dimensions is smaller than the synthesis rate in d = 3dimensions. This is because diffusive currents are smaller in lower dimensions and hence ribosomes are more depletedat the filament entrance. For small values of D eff , the current is proportional to D eff , namely, J = α ∞ µ d D eff + O ( D ) , (26)where the proportionality constant is the ratio between the entry rate α ∞ for circularized mRNA and the constant µ d that depends on the geometry of the problem.
3. Two-dimensional rectangular box
We consider the case of a filament immersed into a medium that has the shape of a two-dimensional rectangularbox. We assume that the box is centered at the origin r = 0 and that the sides of the box have lengths L x and L y .We derive an explicit expression for the Green function in a two-dimensional rectangular box with the method ofimages [58]. The Green function of a point source in a two-dimensional rectangular box is identical to a series ofGreen functions in R associated with images of the point source, namely, it holds that G L x ,L y ( r , r (cid:48) ) = G ( r , r (cid:48) ) + (cid:88) j ∈N G ( r , r ( j ) ) , (27) theory, d = 2theory, d = 3simulation, d = 3simulation, d = 2 theory, d = 2theory, d = 3simulation, d = 2simulation, d = 3 J/p J/pd αβ d αβ (a) (b) FIG. 4: Protein synthesis rate
J/p as a function of the filament end-to-end distance d αβ for parameters α ∞ /p = 0 . β/p = 1for D eff = 1 [Panel(a)] and D eff = 0 . R ( d = 2, red solid lines) and R ( d = 3, black dashed lines) are compared with simulations results for filaments consisting of (cid:96) = 100 monomers (markers).The theoretical result Eq. (12) applies for d αβ > J = 0 .
24 for d αβ = 0. Therefore, we have added dotted lines connecting J = 0 .
24 for d αβ = 0 with J at d αβ = 2. The remaining parameters that specify the simulations can be found in Sec. IV. where r ( j ) are the coordinates for the images of the point source located at r (cid:48) , see Fig. 3 for an example, and G isthe Green function in Eq. (21).Substituting the Green function given by Eq. (27) in Eq. (20), we obtain the expression Eq. (8), with now µ ( L x , L y ) = 1 + log d αβ + I L x ,L y , (28)and where I L x ,L y is the series I L x ,L y = (cid:88) j ∈N β log | r α − r ( j ) β | − (cid:88) j ∈N α log | r α − r ( j ) α | . (29)The sums in Eq. (29) run over the images of the initiation and termination sites of the filament, which defines the set N α and N β . The specific locations of r ( j ) α and r ( j ) β are detailed in Fig. 3. As shown in Ref. [35], the series Eq. (29)converges rapidly since the influence of the copies r ( j ) α and r ( j ) β on the concentration of ribosomes in the original systemdecreases fast enough with the distance.Lastly, we note that the method of images works for a rectangular shaped box since two-dimensional Euclideanspace can be tiled with rectangles. Other geometrical shapes that allow for a complete tiling of space are trianglesand hexagons, see [35] and references therein.
4. Three-dimensional cuboid
An analytical expression for the protein synthesis rate can also be derived in the case of a three-dimensional cuboidwith linear dimensions L x , L y and L z . We then obtain formula Eq. (8) with µ ( L x , L y , L z ) = 25 − (cid:18) d αβ + I L x ,L y ,L z (cid:19) (30)where I L x ,L y ,L z is the series I L x ,L y ,L z = (cid:88) j ∈N β | r α − r ( j ) β | − (cid:88) j ∈N α | r α − r ( j ) α | . (31)The sums run over the images of the initiation and termination sites of the filament in R . For more in-depth analysisof the finite volume effects, see Ref. [35]. theory, d = 2theory, d = 3simulation, d = 3simulation, d = 2 J/p D eff FIG. 5: Protein synthesis rate
J/p as a function of the effective diffusion contant D eff for filaments in R ( d = 2) and R ( d = 3).Analytical results from mean-field theory [solid lines depicting Eq. (12) with µ d as in Eqs. (22) or (25)] are compared withsimulation results (circles). The parameters used to compute the theoretical curves are d αβ = 20, α ∞ /p = 0 .
4, and β/p > / D eff →∞ J/p = 0 . IV. COMPARING MEAN-FIELD THEORY WITH SIMULATIONS
We have performed numerical simulations of the RTD to check the mean field assumptions in Eq.(5), as shownin Figs. 4 and 5. Theory and simulations are in very good correspondence, despite the fact that theory neglectscorrelations between particles, finite size effects on the filament due to boundary layers, and finite size effects dueto the finite volume of the reservoir. The very good correspondence between numerical experiments and theorydemonstrates that the expression for the current J given by Eqs. (10-12) is useful to quantify how finite mobilityaffects the protein synthesis rate J .In what follows, we detail the specifics of the Monte Carlo simulations. Both components of the RTD, i.e., diffusionof particles and the active transport on the filament, can be simulated independently using a continuous-time MonteCarlo simulation on the TASEP [54, 55] and a Brownian motion in the reservoir. However, in order to simulate theRTD model, we need to couple the dynamics of the two processes. A. Monte Carlo simulations of the RTD
In this subsection, we describe the algorithm used to simulate the dynamics of ribosome (i) in the reservoir, (ii)on the filament and (iii) how these two subsystems are coupled at the first and last site of the filament where theribosomes respectively enter on and exit from the filament.First, we detail the simulations of the unbound ribosomes diffusing in the reservoir. We consider that unboundribosomes do not interact with each other and their position (cid:126)r evolves according to a Brownian equation of motion d(cid:126)rdt = (cid:126)ξ ( t ) , (32)where (cid:126)ξ is a white noise such that (cid:104) ξ a ( t ) (cid:105) = 0 , (33) (cid:104) ξ a ( t ) · ξ b ( t (cid:48) ) (cid:105) = 2 Dδ ( t − t (cid:48) ) δ a,b , (34)where the indices a and b stand for the space coordinates, i.e., x and y for a two-dimensional reservoir; x , y and z for three-dimensional reservoir. We integrate these equations numerically by discretizing time into intervals of length∆ t = t − t (cid:48) , such that (cid:126)r ( t + ∆ t ) − (cid:126)r ( t )∆ t = (cid:126)ξ ( t ) . (35)The δ ( t − t (cid:48) ) in the amplitudes of the white noise are replaced by 1 / ∆ t , leading to the following update for each spacecomponent r a ( t + ∆ t ) = r a ( t ) + √ D ∆ t ξ a , (36)0Second, we detail the simulations of ribosomes bound to a filament located inside the reservoir. The filamentcontains (cid:96) sites and each site has the length r of a ribosome. The filament has thus a total length L = (cid:96)r . Thedynamics on the TASEP is performed with a continuous-time Monte Carlo algorithm [54, 55], sometimes calledGillespie algorithm [53]. The current configuration of ribosomes on the filament allows only a finite number of movesof ribosomes each given by the TASEP rules described above. For illustration, in the particular case of Fig.1, thefirst site is empty, thus a ribosome can enter at a rate α = ˜ αN ( t ); three ribosomes are present in the bulk of thefilament without another ribosomes on their right side, thus they can jump to the right at a rate p ; and finally aribosome occupies the exit site of the filament, so it can leave at the rate β the filament and return to the reservoirto resume a Brownian motion. It is useful to define the sum S r of the possible transition rates; in the case of Fig.1, S r = α + β + 3 p . A particular move is chosen with a probability linearly related to its rate and the filament is forcedto perform this move. For instance, the probability P β to move the ribosome from the exit site to the reservoir is P β = β/S r . This algorithm thus avoid rejection of ribosome moves, which spares a lot of computational time inthe case of low and high densities of ribosomes, compared to a sequential algorithm. Indeed, a sequential algorithmconsists in choosing randomly a site of the filament. This site is likely empty in the LD phase or stuck into a jam inthe HD phase, leading in both cases to frequent rejections of the move trial. The other advantage of a continuous-timeMonte Carlo is that the time of evolution of the filament during this move is explicitly defined from the transitionrates like τ ∼ S − r , and can thus take continuous values. The explicit definition of τ will be useful to couple the timescale of the dynamics between the ribosomes on the filament and the ribosomes performing Brownian motion in thereservoir. Note that, intuitively, the sum of rate S r , and thus the time τ spent by the filament during a move bothdepend on the configuration of ribosomes. If S r is small (large), i.e., if a transition is unlikely (resp. likely) to happen,then the time evolution of the filament will be large (resp. small).Third, we discuss how the dynamics in the reservoir is coupled to transport on the filament. First we draw atime τ from the continuous-time Monte Carlo algorithm, then update the reservoir configuration over this time byintegrating the Brownian equations for each particle in the reservoir over the time τ , and then draw another time τ and so on. Hence, in this approach, we assume that in the time τ the reservoir does not change significantly. Theinternal dynamics of ribosome hopping is by definition not coupled to the reservoir as, in the RTD the ribosomes canneither attach nor detach in the bulk of the filament. The coupling between reservoir and filament takes place atthe first and last site of the filament. Therefore, it is sufficient to define the positions r α and r β of the first and thelast sites, respectively. Note that the total length L , which may however be different than the end-to-end distance d αβ = | r β − r α | , which can take any value between 0 and L depending on the conformation of the filament. Amongthe possible moves accounted in the simulation is the attachment of a ribosome at the entrance: we define a sphericalreaction volume V α = 4 / πr of radius r centered at the first site of the TASEP. If a ribosome in the reservoir ispresent in V α and if the first site is empty, then it can attach at a rate ˜ α (define in Eq.(1)). In the same way, aspherical volume V β of radius r is centered at the exit site of the filament. If a ribosome exits the filament at a rate β , then it is released at a random position inside V β and resumes a Brownian motion in the reservoir. Note that wehave used the same numerical technique successfully in Ref. [19] to coupled the TASEP-LK with Brownian particlesinside a reservoir. B. Parameters of the simulations
We describe in this paragraph the geometry of the simulated RTD. First, the filament is chosen to have a totalcontour length L = (cid:96) r with (cid:96) = 100, which is enough to keep boundary effects on the TASEP of the order of a fewpercents on the exact current, i.e., of the order of statistical fluctuations [59, 60]. In the simulations, the filament islocated in the middle of the reservoir to ensure isotropy of the particle concentration and limiting boundary effects.Second, the reservoir is chosen to be large with respect to d α,β . In three dimensions, we choose the dimensions L x = L y = 100 r in the orthogonal direction to the end-to-end distance, whereas the longitudinal direction to d α,β is taken to be larger, i.e., L z = 200 r . In two dimensions, we choose L x = 400 r in the longitudinal direction and L y = 200 r in the orthogonal direction. Note that the gradient of ribosomes in the reservoir induced by the transporton the filament is expected to be larger along the longitudinal direction to d α,β . This is why this dimension is chosenlarger than the orthogonal directions. With these reservoir dimensions, boundary effects are small as the system islarge with respect to the gradient of particles. The reflecting boundary conditions are implemented as follows: if theupdate of a Brownian particle leads to a position outside of the box, the move is rejected.We now discuss the remaining parameters of the system linked to the concentration of ribosomes, attachmentrate at the entry site of the filament and the diffusion coefficient of the Brownian motion. In three dimensions, wechoose to include 10 ribosomes in the reservoir, leading to a density of ribosomes c ∞ = 0 . r − ; whereas in twodimensions, we choose 5 . ribosomes in the system, leading to a density 6 . r − . In two and three dimensions,we choose ˜ α = 0 . /c ∞ , so that α ∞ = ˜ α c ∞ = 0 . D = 0 . αr and D = ˜ αr in Fig.3(a) and (b), so that D eff = D/ (˜ αr ) = 0 . β = p = 1 like the analytical calculation, i.e., all rates1can be seen to be expressed in unit of p .In this paragraph, we discuss how we choose correlation and equilibration time to increase the quality of thesampling during Monte Carlo simulations. The correlation time can be approximated by the time needed to replaceall the ribosomes on the TASEP. During one MC iteration, the time spent during the update is τ ∼ /S r ∼ / ( ρ(cid:96)p )where ρ is the global density of ribosomes on the filament, i.e., ρ = N r /(cid:96) where N r is the total number of ribosomeson the filament. Note that, in the last approximation of τ , the sum of the rates S r is obtained assuming that it isdominated by the hopping rates in the bulk of the TASEP, which contains ≈ ρ(cid:96) particles. The last ribosome thatentered the TASEP will need at least to be chosen (cid:96) times amongst ρ(cid:96) possibilities of moves. Therefore the correlationtime becomes τ c ≈ ρ(cid:96) τ = (cid:96)/p . As p = 1 and (cid:96) = 100 in our simulations (both two and three dimensions), weuse τ c = 100. Starting with an empty initial configuration, we ensure the steady state by performing 100 τ c = 10 iterations described above (continuous time on the filament and integration of the Brownian motion in the reservoir).Subsequently, 2 . samplings are in three dimensions and 10 samplings in two dimensions, each spaced by τ c = 100iterations to decorrelate the configurations. V. BIOLOGICAL RELEVANCE OF DIFFUSION IN RIBOSOMAL RECYCLING
To determine the biological relevance of finite mobility for ribosomal recycling, we use experimentally measuredvalues for the parameters that appear in the theoretical expression for the protein synthesis rate derived in Sec. III. Wefocus on two organisms for which the required microscopic parameters have been measured experimentally, namely,the bacterium
Escherichia coli and the budding yeast
Saccharomyces cerevisiae . Moreover, we focus on the three-dimensional case corresponding to cytoplasmic translation.
TABLE I: Impact of finite mobilities on ribosomal recycling in two organisms.
E. coli S. cerevisiae µ /D eff < . µ /D eff < . Since for physiological parameters the initiation of translation is the rate limiting step, we use the expression forthe protein synthesis rate given by Eq. (13). Eq. (13) implies that if µ d (cid:28) D eff (37)then diffusion has no meaningful influence on the protein synthesis rate. On the other hand, when µ d (cid:29) D eff (38)then the influence of finite diffusion on protein synthesis rate is sizeable. Hence, in what follows we estimate theparameters µ d and D eff . A. Estimate of µ First, we estimate the geometric parameter µ corresponding to cytoplasmic translation. Formula (25), implies fora three dimensional and infinitely large reservoir that µ ≤ , (39)where the equality is achieved in the limit d αβ → ∞ . B. Estimate for D eff in Escherichia coli In order to estimate D eff , it is useful to rewrite the expression Eq. (9) in terms of (cid:104) α (cid:105) , which gives D eff = D (cid:104) N r (cid:105)(cid:104) α (cid:105) r = 4 π Dc u r (cid:104) α (cid:105) , (40)2where c u is the concentration of unbound ribosomes. The quantity (cid:104) α (cid:105) is hard to estimate since it can vary in severalorders of magnitude from one mRNA transcript to another, see for instance Ref. [5]. However, since initiation is therate limiting step, it holds that (cid:104) α (cid:105) < p , (41)with the elongation rate p being fairly independent of the mRNA transcript and the biological organism under study.Combining Eqs. (40) and (41), we obtain the lower bound D eff > π Dc u rp . (42)We are left to estimate the parameters D , c u , r and p . We first consider the case of the bacteria Escherichia coli .Empirical values for the diffusion of ribosomes in
E. coli show that D ≈ . µm /s , see Table 4-1 in Ref. [37].However, the diffusion coefficient of the subunits of unbound ribosomes (i.e., those not bound to mRNA), is one orderof magnitude larger and given by D ≈ . µm /s , as shown in Ref. [43].For the radius of the reaction volume r , we use that the reaction volume cannot be smaller than the radius of aribosome (or one of its subunits), and thus r > E. coli , the elongation rate p has been measured in several experiments, see Refs. [44–46], leading to a value p of about about 10 −
20 codons per second. Since a ribosome occupies three codons, we take for p ≈ s − .Lastly, we need an estimate for the concentration c u = N u V . (43)The volume of
E. coli is V ≈ µm and its total number of ribosomes is about N tot = 20000 [37]. The fraction ofunbound (or free) ribosomes is about 15% [43, 47] of the total value, leading to c u ≈ × . × µm − ≈ × µm − . (44)Combining all parameter values into the right hand side of the bound Eq. (42) for D eff , we obtain that D eff > π . × × × µm ≈ . , (45)and therefore µ D eff < . . (46)We can conclude that diffusion has no sizeable effect on protein synthesis rates. This is in particular true sincewe have been very generous with all the biological parameters. For example, taking (cid:104) α (cid:105) < p/
20 instead of p/
2, as inRef. [48], would provide an even smaller upper bound µ D eff < . C. Estimate for D eff in Saccharomyces cerevisiae As a second example, we consider the case of budding yeast. We use again Eq. (42) to bound D eff . All empiricalvalues are known for this organism, see for instance table S1 in Ref. [5].Empirical values for the diffusion coefficient of the 60S subunit of ribosomes in the dense nucleoplasm of buddingyeast show that D ≈ . µm ) /s [49]. We may expect that ribosomes diffuse faster in the cytoplasm, where translationtakes place.For the radius of the reaction volume r , we use again the reaction volume cannot be smaller than the radius of theribosome, and thus r > p ∼
10 codons per second and therefore p ≈ s − since a ribosome occupies three codons [5, 50].Finally, we come to the estimate of c u , given by Eq. (43). The volume of a budding yeast cell is about V ≈ µm [5, 51] and the number of ribosomes is 2 × [5, 30, 52]. Using again that a fraction 15% of ribosomes are unbound,see Figure 3 in [5], we obtain c u ≈ × . × µm − ≈ × µm − , (47)3which is in fact close to the concentration of unbound ribosomes in E. coli , see Eq. (44).Combining all parameters in the bound given by Eq. (42), we obtain that D eff > π . × × × µm ≈
59 (48)and µ D eff < . . (49)We should again bear in mind that the bound in Eq. (49) is a generous upper bound based on the bound on theinitiation rates given by Eq. (41), and it is thus likely a loose bound and a significant overestimate for µ /D eff . VI. DISCUSSION
We have made a study of a totally asymmetric simple exclusion process immersed in a diffusive reservoir [11, 26],which we have called the RTD model. The RTD is a model for translation based on directed transport of ribosomesalong mRNA and recycling of ribosomes through diffusion in the cytoplasm. We have used this model to determinewhether under physiological conditions finite diffusion is a limiting factor for ribosome recycling.We have derived an analytical expression for the current J at which mRNA is translated into proteins, which iscorroborated by numerical simulation results. These results show that that finite diffusion leads to a reduction in thetranslation rate J because the concentration of ribosomes at the mRNA initiation site is depleted. In addition, wefind that the ratio between a geometric parameter µ d and an effective diffusion coefficient D eff determines whetherdiffusion has an impact on the protein synthesis rate: if µ d (cid:28) D eff , then the concentration of ribosomes at the 5’ endof the mRNA is not affected by finite diffusion; on the other hand, if µ d (cid:29) D eff then depletion of ribosomes at themRNA initiation site is significant.Using a broad range of physical parameters, we find that it is unlikely that finite diffusion is a limiting factorunder physiological conditions in ribosome recycling. Indeed, in Table I, we present generous upper bounds for theparameter µ d /D eff for two organisms, namely, the bacterium E. coli and the yeast
S. cerevisiae . In both cases, weobtain that µ d /D eff is substantially smaller than 1.The outcome of our analysis, namely that the finite mobility of ribosomes does not play a role in translation control,is not a complete surprise, given that ribosomes diffuse at large enough rates. For example, it takes 0 . E. coli cell and 10 s for a protein to diffuse across a yeast cell [39], while the time to translatea protein is about 2 min [39]. Hence, as much as concerns the translation of mRNA into proteins, the diffusion rateof ribosomes can be considered very large and therefore of negligeable effect on the whole translation process. Also,since ribosomes biogenesis is one of the most resource expensive process for the cell [29, 30], it is reasonable to assumethat the molecular conditions are optimised by evolutionary constraints in order to render translation efficient, whichin the present context implies that translation is not limited by ribosome mobilities.From a biological point of view, these results imply that mRNA circularisation [1, 9] is not due to optimizing therecycling of ribosomes through diffusion in the cytoplasm. Instead, the circularisation of mRNA may regulate theefficiency of translation initiation through the binding strength of initiation factors to the mRNA [9, 10]. Hence,we come to a different conclusion than Ref. [7], which argues that three-dimensional diffusion of ribosomes in thecytoplasm plays an important role for mRNA translation control. Note that the question of the effect of the finitemobility of ribosomes on the current on mRNA remains open in two dimensions, as the diffusion coefficient of ribosomesconstrained to a two-dimensional diffusion on the endoplasmic reticulum is not known to our knowledge.Although finite diffusion is not limiting for ribosome recycling under physiological conditions, the RTD model may berelevant to explain the reduction in protein production when cells are in a dormant state. The mobility of cytoplasmicparticles in dormant yeast cells is much lower than their mobility in yeast cells under normal conditions [31, 32]. Thereduction in mobility of cytoplasmic particles is due to a transition between a fluid-like to a solid-like phase of thecytoplasm, which is triggered by the acidification of the cytosol [32]. The formula J ∼ D indicates that the proteinsynthesis rate will scales proportional to the particle mobility.The RTD model is also interesting as a model for the coupling between active transport and passive diffusion.Remarkably, the rate J admits a universal form that depends on five parameters only: the elongation rate p , theratio β/p between the rate β of termination and p , the ratio α ∞ /p between the initiation rate α ∞ for a homogeneousreservoir (i.e., the limit of an infinitively fast diffusion) and p , an effective diffusion constant D eff , and a dimensionlessparameter µ d that quantifies the effect of the geometry of the reservoir and the filament on the current J . We have alsofound an interesting qualitative distinction between finite diffusion in two and three dimensions. In two dimensions,it holds that the current J vanishes in the large distance (between sink and source) limit, while in three dimensions4this limit gives a finite current J . However, the decay towards zero of J in two dimensions, which may be relevantfor the endoplasmic reticulum translation, is logarithmically slow.We end the paper by discussing the assumptions made by the RTD model and interesting future extensions of thepresent paper. First, we have ignored the fact that ribosomes disassemble into two subunits in the cytoplasm [1].Hence, in principle we should consider a reservoir with two types of particles. However, if the mRNA binding rate oneof these subunits is rate limiting, then the predictions of our model would remain valid. Interestingly, experimentaldata indicates that in prokaryotes the binding of the 40S ribosomal subunit is thought to be the rate-limiting stepof initiation [4]. Second, we have assumed that mRNA has zero mobility and we have also assumed that the end-points of the mRNA are immobile. However, including diffusion of the mRNA in the model would not alter the mainconclusions of this paper, since it would only reduce the effects of finite diffusion on the protein synthesis rate. Third,it is known that cytoplasmic particles diffuse anomalously within living cells [40–42] and therefore a model based onfractional Brownian motion is more appropriate [42]. However, the exponent of the anomalous diffusion is close to 1(0 .
88 for nanosilica particles of various sizes in yeast cells [32]), and therefore we expect it not to have a major impacton short length scales. It would nevertheless be interesting to analyse the dependence of J on d αβ in this case. Acknowledgments
This work was supported in part by a Mod´elisation pour le Vivant CNRS Grant CoilChrom (2019–2020), and theLabEx NUMEV (ANR-10-LABX-0020) within the I-SITE MUSE of Montpellier University [No. AAP 2013-2-005,and Flagship Project Gene Expression Modeling (2017–2020)].
Appendix A: Concentration of ribosomes in the box
We solve Eqs. (16)-(17) in various geometries when | r β − r α | > r .In R , we obtain that c ( r ) = c ∞ + Jr D V − J D V | r − r β | + Jr D V ln (cid:16) | r − r α | r (cid:17) , | r − r β | < r,c ∞ − Jr D V + J D V | r − r α | − Jr D V ln (cid:16) | r − r β | r (cid:17) , | r − r α | < r,c ∞ − Jr D V ln (cid:16) | r − r β || r − r α | (cid:17) , | r − r α | > r, and | r − r β | > r, (A1)while in R it holds that c ( r ) = c ∞ + Jr D V − J D V | r − r β | − Jr D V | r − r α | , | r − r β | < r,c ∞ − Jr D V + J D V | r − r α | + Jr D V | r − r β | , | r − r α | < r,c ∞ + Jr D V (cid:16) | r − r β | − | r − r α | (cid:17) , | r − r α | > r, and | r − r β | > r. (A2)For a rectangular box of dimensions L x × L y , we obtain c ( r ) = c ∞ + Jr D V − J D V | r − r β | + Jr D V ln (cid:16) | r − r α | r (cid:17) + c I ( r ) , | r − r β | < r,c ∞ − Jr D V + J D V | r − r α | − Jr D V ln (cid:16) | r − r β | r (cid:17) + c I ( r ) , | r − r α | < r,c ∞ − Jr D V ln (cid:16) | r − r β || r − r α | (cid:17) + c I ( r ) , | r − r α | > r, and | r − r β | > r. (A3)where c I ( r ) = Jr D V (cid:88) j ∈N α ln( | r − r ( j ) α | ) − Jr D V (cid:88) j ∈N β ln( | r − r ( j ) β | ) , (A4)and where the r ( j ) α denote the coordinates of the images of the filament initiation site located at r α , and where the r ( j ) β denote the coordinates of the images of the filament termination site r β , as illustrated in Figure 3.Analogously, for a cuboid of dimensions L x × L y × L z , we obtain c ( r ) = c ∞ + Jr D V − J D V | r − r β | − Jr D V | r − r α | + c I ( r ) , | r − r β | < r,c ∞ − Jr D V + J D V | r − r α | + Jr D V | r − r β | + c I ( r ) , | r − r α | < r,c ∞ + Jr D V (cid:16) | r − r β | − | r − r α | (cid:17) + c I ( r ) , | r − r α | > r, and | r − r β | > r, (A5)5where c I ( r ) = − Jr D V (cid:88) j ∈N α | r − r ( j ) α | + Jr D V (cid:88) j ∈N β | r − r ( j ) β | . (A6) [1] H. Lodish, et al., Molecular cell biology , eighth edition, (W.H. Freeman and Company, 2016).[2] M. Kosak,
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