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Ribosome Recycling, Diffusion, and mRNA Loop Formation in Translational Regulation

Department of Biomathematics, and Institute for Pure and Applied Mathematics, University of California at Los Angeles, Los Angeles, California

Correspondence: Address reprint requests to Tom Chou, Dept. of Biomathematics, Box 951766, UCLA, Los Angeles, CA 90095-1766. Tel.: 310-206-2787; Fax: 310-825-8685; E-mail: tomchou{at}ucla.edu.

	ABSTRACT

TOP ABSTRACT INTRODUCTION PHYSICAL MODELS RESULTS AND DISCUSSION EXPERIMENTAL CONSEQUENCES AND... SUMMARY APPENDIX A: PHYSICAL ASSUMPTIONS... APPENDIX B: MEAN FIELD... APPENDIX C: OPEN CHAIN... APPENDIX D: ASYMPTOTICS FOR... ACKNOWLEDGEMENTS REFERENCES

 
We explore and quantify the physical and biochemical mechanisms that may be relevant in the regulation of translation. After elongation and detachment from the 3' termination site of mRNA, parts of the ribosome machinery can diffuse back to the initiation site, especially if it is held nearby, enhancing overall translation rates. The elongation steps of the mRNA-bound ribosomes are modeled using exact and asymptotic results of the totally asymmetric exclusion process. Since the ribosome injection rates of the totally asymmetric exclusion process depend on the local concentrations at the initiation site, a source of ribosomes emanating from the termination end can feed back to the initiation site, leading to a self-consistent set of equations for the steady-state ribosome throughput. Additional mRNA binding factors can also promote loop formation, or cyclization, bringing the initiation and termination sites into close proximity. The probability distribution of the distance between the initiation and termination sites is described using simple noninteracting polymer models. We find that the initiation, or initial ribosome adsorption binding required for maximal throughput, can vary dramatically depending on certain values of the bulk ribosome concentration and diffusion constant. If cooperative interactions among the loop-promoting proteins and the initiation/termination sites are considered, the throughput can be further regulated in a nonmonotonic manner. Experiments that can potentially test the hypothesized physical mechanisms are discussed.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION PHYSICAL MODELS RESULTS AND DISCUSSION EXPERIMENTAL CONSEQUENCES AND... SUMMARY APPENDIX A: PHYSICAL ASSUMPTIONS... APPENDIX B: MEAN FIELD... APPENDIX C: OPEN CHAIN... APPENDIX D: ASYMPTOTICS FOR... ACKNOWLEDGEMENTS REFERENCES

 
The rate of protein production needs to be constantly regulated for all life processes. Genetic expression, protein production, and post-translational modification, as well as transport and activation, are all processes that can regulate the amount of active protein/enzymes in a cell. Although much recent research has focused on the biochemical steps regulating the switching of genes and rates of transcription, translational control mechanisms, post-translational processing, and macromolecular transport are also important. For example, during embryogenesis, nuclear material is highly condensed, transcriptional regulation is inactive, and translational control is important (Browder et al., 1991; Wickens et al., 1996). In other instances, transcriptional regulation is accompanied by long lag times, particularly with long genes. Translational regulation is also the only means by which RNA viruses express themselves.

Protein production, as with other cellular processes, requires the assembly of numerous specific enzymes and co-factors for initiation. This assembly occurs in the cytoplasm and on the 5' initiation site of mRNA. Translation involves unidirectional motion of the ribosome complex along the mRNA strand as amino-acid-carrying tRNA successively transfer amino acids to the growing polypeptide chain. Images of mRNA caught in the act of translation often show numerous ribosome complexes attached to the single-stranded nucleotide (Fig. 1 A). The multiple occupancy is presumably a consequence of very active translation, when many copies of protein are desired.

View larger version (84K): [in this window] [in a new window]   FIGURE 1  (A) An electron micrograph of polysomes on mRNA. (B) An AFM micrograph of circularization of mRNA mediated by loop forming proteins. From Wells et al. (1998). These images are of double-stranded RNA of approximate length 2–4x the dsRNA persistence length. Single-stranded end segments with loop binding factors comprise the ends.

Even in uncapped mRNA, there is evidence that certain sequences in the terminal 3' untranslated region (UTR) can enhance translation to levels comparable to those seen in capped mRNAs (Wang et al., 1997; Jackson, 1996). Additionally, there are indications that proteins near the termination end can, upon contact, directly activate (Gallie, 1991) or inactivate (Curtis et al., 1995; Dubnau and Struhl, 1996) ribosome entry at the 5' initiation site. Loops also appear to be a common motif in DNA structures (Goddard et al., 2000; Zacharias and Hagerman, 1996) and take part in transcriptional regulation (Dunn et al., 1984; Wyman et al., 1997). Double-stranded DNA has a much longer persistence length than single-stranded nucleic acids (such as mRNA) and is much less likely to form loops without accompanying binding proteins or specific sequences. Direct evidence for RNA circularization is shown in Fig. 1 B, which shows loop formation of relatively short double-stranded mRNA in the presence of loop-binding factors at their ends (Hagerman, 1985). It is reasonable to expect that the more flexible single-stranded mRNA decorated with ribosomes can form similar loops. Besides the AFM-imaged loop of double-stranded RNA shown in Fig. 1 B, there is also substantial evidence, particularly in viral mRNAs, that basepairing between uncapped 5' regions and nonpolyadenylated 3' regions forms closed loops of many kilobases (Wang et al., 1997). This loop formation by direct basepairing, or "kissing," is a very plausible mechanism by which the 3' UTR recruits ribosomes and delivers them to the 5' initiation site (Guo et al., 2001).

In this article, we model the proposed cyclization, i.e., "circularization" (Sachs et al., 1997), and ribosome recycling mechanisms. Cooperative interactions of the initiation and termination sites with eukaryotic initiation factors (eIFs) and PAB proteins will also be considered within a number of reasonable assumptions. Since translation employs an immense diversity of mechanisms and proteins that vary greatly across organisms (Mathews et al., 1996), we will only develop an initial, qualitative physical picture of cytoplasmic mRNA translation consistent with the ingredients mentioned above. Three different coupled effects are considered in turn: 1), a totally asymmetric exclusion process (TASEP) describing the unidirectional stochastic motion of the ribosome along the mRNA; 2), the diffusion and adsorption/desorption kinetics from the mRNA initiation/termination sites; and 3), the polymer physics associated with how the termination and initiation sites are spatially distributed relative to each other. The ribosome density along the mRNA, as well as the time-averaged throughput of ribosomes, i.e., the ribosome "current," are described by solutions of the TASEP. The parameters in the TASEP are the internal hopping rates and the injection and extraction rates at the initiation and termination sites, respectively. Since ribosome components that diffuse in bulk must adsorb on the initiation site, the injection rate used in the TASEP will be proportional to the local concentration of the rate-limiting ribosome. Ribosomes that reach the termination site desorb and reenter the pool of diffusing ribosomes. The distance between the termination end and the initiation site, when ribosomes are released, can thus influence the absorption rate and hence the overall translation rate. The initiation-termination end-to-end distance distribution can be estimated with basic polymer physics. The end-to-end distance distribution can include effects such as specific binding of poly(A)-associated proteins with the 5' cap, thereby forming a loop, bringing the initiation and termination sites into close proximity. Although our model applies only to cytoplasmic mRNA translation, many of its components can also be adapted to treat mRNA adsorption on endoplasmic reticulum (ER) and ER-assisted translation.

	PHYSICAL MODELS

TOP ABSTRACT INTRODUCTION PHYSICAL MODELS RESULTS AND DISCUSSION EXPERIMENTAL CONSEQUENCES AND... SUMMARY APPENDIX A: PHYSICAL ASSUMPTIONS... APPENDIX B: MEAN FIELD... APPENDIX C: OPEN CHAIN... APPENDIX D: ASYMPTOTICS FOR... ACKNOWLEDGEMENTS REFERENCES

 
We now consider the physical processes necessary to describe the above-mentioned translation processes. At the relevant timescales, we will see that fluctuations in these physical mechanisms are uncorrelated with each other. This allows us to consider simple steady states where time or ensemble averages of the TASEP, ribosome diffusion in the cytoplasm, and the mRNA chain conformations are uncorrelated and can be taken independently of each other. A simplifying schematic of the basic ingredients of mRNA translation is given in Fig. 2 .

View larger version (14K): [in this window] [in a new window]   FIGURE 2  A cartoon of mRNA translation in eukaryotes. The intermediary proteins and co-factors are not depicted. (A) An mRNA chain loaded with ribosomes (green), in various stages of protein (black) production. Ribosomal components as well as other components such as tRNA exist at a uniform background concentration. The initiation and termination sites are additional sinks (i = 1) and sources (i = N), respectively, of ribosomes. (B) Binding factors (yellow and dark gray) can increase the probability of loop formation or circularization, which brings the poly(A) tail (red) in better proximity to the initiation site, enhancing ribosome recycling. (C) Schematic of the associated TASEP with injection (), internal hop (p), and desorption (ß) rates labeled.

(1)

where

(2)

In the N limit, the one-dimensional TASEP (Eq. 1) admits three nonequilibrium steady-state phases, representing different regimes of the steady-state current J:

(3)

The phases I, II, and III defined by Eq. 3 are denoted as the maximal current, low density, and high density phases, respectively, and are delineated in Fig. 3 by the dotted phase boundaries. Qualitatively, when ß is small, and injection rates are faster than extraction rates ( > ß), the rate-limiting process is the exit step at i = N. Therefore, the high occupancy phase II has a low current which is a function of only the slow step ß. In the opposite limit of fast desorption at i = N, and slow injection at i = 1 (small ), the chain is always nearly empty, and has a small current J that depends only upon the rate-limiting step . For large ß, the system attains maximal current J = p/4 where the effective rate-limiting steps are internal hopping rates p. In this phase, the constant current J = p/4 is independent of further increases in or ß. The ribosomal currents given by Eq. 3 and the associated phase diagram in Fig. 3 are valid only in the N limit. Nonetheless, the N = phase diagram is qualitatively accurate for the currents expected at large but finite N.

View larger version (16K): [in this window] [in a new window]   FIGURE 3  The infinite chain (N ) limit nonequilibrium phase diagram of the standard TASEP. The maximal current (III), low density (I), and high density (II) phases and their corresponding steady-state currents are indicated. In this and subsequent phase diagrams, solid curves correspond to phase boundaries across which the slope of the steady-state currents (with respect to the parameters) is discontinuous. Across the dashed phase boundaries, the currents and their first derivatives are continuous.

What remains is to determine the self-consistent dependence of the model parameters, in particular and ß, on the local ribosome concentration (which in turn depends on the mean current J), diffusion rates, circularization, etc. For example, the injection rate at the initiation site will be proportional to a microscopic binding rate k x the local ribosome concentration.

Steady-state release, diffusion, and capture
The complete mRNA translation machinery is extremely complicated, since it is comprised of many auxiliary RNA and protein co-factors, as well as a collection of active mRNA chains. Since there are many active mRNA chains in the cytoplasm, each mRNA chain feels the sinks (initiation sites) and sources (termination sites) of all the other mRNA chains. However, these other randomly distributed chains, each with their own initiation and termination sites, contribute an averaged background ribosome concentration. Thus, it is only the termination site (ribosome source) associated with the initiation site on the same mRNA chain that resupplies the initiation site in a correlated manner. We thus consider a single isolated mRNA chain and for the sake of simplicity, assume that a single component, say phosphorylated elongation initiation factor eIF4F or eIF2, say (Clemens, 1996; Sachs and Varani, 2000), is key to a rate-limiting step. We will generically call this component the ribosome. Consider a source of newly-detached ribosomes (emanating from the 3' termination site) at position r away from the 5' initiation site. The probability of finding this particle within the volume element dr about r obeys the linear diffusion equation with the termination site acting as a source,

(4)

where D is the bulk ribosome diffusion constant, J(t) is the instantaneous rate of ribosome release from the termination end, and W_eff(r)dr is the probability that the termination site is within the positions r and r + dr from the initiation site. Although Eq. 4 can be solved exactly for all times, the TASEP result (Eq. 1) is appropriate only in the steady state, so we must consider that limit for all processes.

The typical mRNA passage time of a single ribosome is on the order of 1 min. The bulk diffusion constant of the 10- to 15-nm radius (a 15 nm) ribosome unit is D 10^-8 - 10^-7 cm²/s. A ribosome molecule will diffuse the length of a 1 kB pair mRNA strand in 0.1 s. Therefore, with each release of a ribosome from the termination site, the probability density appears as a pulse which passes through the initiation site over a timescale shorter than it takes for a ribosome to stochastically hop a few lengths of its size along the mRNA chain. Therefore, an upper bound on the amount of correlation between concentration fluctuations and can be found by considering the equal time two-point correlation in the maximal current phase (Derrida and Evans, 1993). Two-point correlations in other current regimes are smaller, and decay exponentially with N (Essler and Rittenberg, 1996). Therefore, we can neglect the correlation of the current J(t) with the occupancy at the initiation site. Moreover, the end-to-end distribution W_eff arises from the statistics of the mRNA polymer configurations and is also assumed independent of both J(t) and The steady-state ribosome distribution can thus be found by setting _tP(r,t) = 0 on the left-hand side of Eq. 4 and taking the time, or ensemble, average of the remaining Poisson equation to obtain

(5)

where J J(t) is the steady-state current of ribosomes emanating from the termination end of the mRNA reentering the bulk ribosome pool, and C(r) = P(r,t) is the ensemble average of P(r).

The boundary condition for C(r) at the initiation site will depend on the occupancy of that site. When it is empty, there is a flux due to the microscopic adsorption step onto the first site. When the bulk ribosome probability distribution will obey perfectly reflecting boundary conditions. Since the probability at r = a, P(r = a, t) depends on the occupation The mean concentration at r = a must be found by averaging the currents in the two states, and When the initiation site is empty,

(6)

Since the steady-state current when the initiation site is full, the averaged steady-state current is (Berg and Ehrenberg, 1983)

(7)

where (1 - ₁) is the fraction of time that the initiation site is unoccupied, ready to absorb a ribosome from the bulk. This probability is not directly dependent on the distribution W_eff(r), but will depend on the time-averaged local concentration C(r), which in turn depends on W_eff only through the distance of the source site at i = N.

The solution to Eq. 5, obeying the boundary conditions Eq. 6 and is

(8)

where r is distance measured from the initiation site, and

(9)

is the associated Green function. In Eq. 9, r_<(r_>) is the smaller(larger) of |r|, |r'| and Y_m() are the spherical harmonic functions of the solid angle defined by the vector r (Arfken, 1985). The first two terms in Eq. 8 arise from the uniform concentration C at infinity and the effects of a sink of radius a at the initiation site. The sink decreases the effective concentration to a level below that of C. The last term proportional to J increases the local concentration and is the result of the source (termination site) some finite distance away from the initiation site. If k , and ribosomes do not bind even when the initiation site is empty, the current J must vanish, and as expected. However, one cannot simply consider the limit k in Eq. 8 because k and ₁ are related through J, the current determined by the TASEP in the rest of the chain. This can be seen by considering the limit k . If the rest of the TASEP contains the rate-limiting step to ribosome throughput, making J very small, it will effectively block clearance of the initiation site, since all sites of the chain will be nearly occupied. In this case, ₁ 1 and k(1 - ₁) is small (despite a large k), and C(r) C, as expected. However, if the rest of the chain is not rate-limiting, and if clearance of the initiation site can occur fast enough, ₁ < 1 and k(1 - ₁) can be large. In this case, The TASEP current J will eventually be balanced with J = (1 - ₁)J₀. Note that J is determined by Eq. 1 which in turn depends on the entry rate (in other words, kC(a)). Thus, steady-state currents need to be self-consistently determined, since C(a) and ₁ are not parameters, but dynamical variables that will in turn be determined by setting J = (1 - ₁)J₀. The analysis which uses Eq. 1 to find self-consistent explicit expressions for J will be presented in the Results and Discussion section.

Since the averaged bulk concentration profile is spherically symmetric about the initiation site, only the = m = 0 terms in the expression for G(r - r') survive and

(10)

where

(11)

The surface concentration at the sink surface a is reduced from the bulk value by a factor of 1 + 4aD/k, due to adsorption and diffusional depletion (Berg and Ehrenberg, 1983). However, part of this initiation site concentration is also replenished at a rate proportional to the flux J, due to the presence of a nearby termination (source) site. The effects of this replenishment are measured by the mean inverse separation 1/R. The harmonic distance, R, defines the effective distance felt by diffusing ribosomes as they make their way from the termination end back to the initiation site. This particular r^-1 scaling is a consequence of the solution to Poisson's equation (Eq. 5) in three dimensions, and is related to the capture probability of diffusing ligands, as analyzed by Berg and Purcell (1977). Equation 10 contains two unknowns, C(a) and ₁. We can use the explicit solution Eq. 1 if we identify the injection rate of the TASEP with the unoccupied initiation site current J₀ = kC(a) . Equation 1 then relates kC(a) to ₁. A second equation can be used by noticing that the flux itself must be balanced. Upon using J = kC(a)(1 - ₁) in Eq. 10, a second relationship between kC(a) and ₁ can be found. Substitution of the solution for kC(a) (in terms of experimentally known or controlled parameters k, C, a, R, and D) into Eq. 1 determines the self-consistent, steady-state ribosome current. This analysis, using the three different explicit forms of Eq. 1 (in the long chain limit) is presented in the Results and Discussion section.

End-to-end distribution W_eff
We now find W_eff(r) to compute R and obtain C(a). In some cases, the mRNA chain may be anchored to cellular scaffolding or ER membranes such that the initiation-termination separation is fixed. If one is interested in steady-state protein production over a period which allows little change in initiation-termination distance, W_eff(r) = (r - R), and R = |R|. In other cases, the mRNA may be free to explore numerous conformations on the protein production timescale. Although it is possible that long mRNA strands may contain secondary structure, we will assume that ribosomes, as they move along the mRNA, melt out these structures. Although there is evidence that mRNA can contain small, local loops (Hagerman, 1985; Wang et al., 1997), it is less likely that they have larger-scale tertiary structure. Thus, we will estimate W_eff and R with simple polymer models.

As shown in Fig. 2, the mRNA is comprised of three segments divided between two qualitatively distinct regions. Typical coding regions are 10³ basepairs, corresponding to N 300. At low ribosome densities, the uncovered mRNA basepairs will be rather flexible, and the effective persistence length will be a local average between a and the 2- to 4-nucleotide persistence length of uncovered mRNA. Large reductions in the persistence length of dsDNA containing segments of single-stranded regions have also been observed by Mills et al. (1994). More sophisticated theories for variable persistence lengths can be straightforwardly incorporated; however, for simplicity, we approximate the persistence length in the coding region to be a uniform constant on the order of = a, the individual ribosome exclusion size. The contour length of the coding region is thus L_N = Na with N 50–500. The untranslated regions, or UTRs between the initiation site and the binding factor (dark gray), and between the termination site and the loop-binding factor (yellow), with persistence lengths , have contour lengths of L_m = m and L_n = n, respectively. Typical L_m, L_n are on the order of 100 bases so that n, m 20–50. However, extremely long noncoding segments of order 1 kbp can exist (Mathews et al., 1996) where m, n 300. In what follows we will also neglect all the excluded volume effects of the remaining short ends of the mRNA chain.

As demonstrated by Wells et al. (1998) in Fig. 1 B, mRNA can form loops in the presence of binding proteins. Therefore, we expect that W_eff(r) (and hence 1/R) will be a linear combination of W(r|open) and W(r|loop), the initiation-termination probability distributions in open and looped mRNA configurations, respectively. These configurations are shown in Fig. 4, A and B. For simplicity, we will use probability distributions associated with noninteracting (phantom) chains and approximate the distributions W(r) with both a freely jointed chain (FJC) and wormlike chain (WLC) models with appropriate persistence lengths . The finite-sized, short distance behavior of the W(r|open, loop) will be important for accurately computing 1/r. As we will see, W(r|loop) can be constructed from the more fundamental quantity W(r|open) (Liverpool and Edwards, 1995; Sokolov, 2002). Since we are eventually interested in either ribosome transport from termination to initiation or in activation/deactivation of initiation or release sites due to direct contact with the end proteins, we compute in Appendix C the distance distribution W(r|open) in the state where site i = N is occupied and site i = 1 is unoccupied.

View larger version (12K): [in this window] [in a new window]   FIGURE 4  A schematic of the effects of loop forming factors. The coding region of the mRNA is blue (the ribosomes and the poly-A tail are not shown), the noncoding spacers of m and n persistence lengths are solid black, while the neglected short ends are dashed curves. The loop binding factors are of typical size d. (A) Nonlooped conformations in which the initiation-termination site distribution function is governed by W(r|open). (B) The initiation-termination distribution function in looped configurations is denoted W(r|loop). W(r|loop) is weighted more strongly at small |r| relative to W(r|open). For stronger attraction between loop binding factors the probability of loop formation increases, decreasing the effective distance R that ribosomes must diffuse to be recycled back to the initiation site.

(12)

where the free energies of a closed and open mRNA chain are G_loop = -U₀ - S_loop and G_open = -S_open, respectively. Since the ratio of the number of configurations under looped and open chain conditions is the ratio of probabilities of loop formation in the absence of head-tail interactions ( and

(13)

The probability, in the absence of loop-binding proteins, that the ends of a noninteracting chain would intersect itself within the interaction volume defined by a thin spherical shell of thickness (the binding interaction range), is approximately

(14)

where d is the typical size of the loop binding factors and We have assumed the total radius of gyration L_T >> a, and used a Gaussian chain as a qualitative approximation for the distributions used in the calculation of The conditional probability distribution W(r|loop) for a looped chain is

(15)

where W(r|open) denotes the single segment, open chain probability distributions in the two segments with persistence lengths = a,. For the loop distribution given by Eq. 15 is qualitatively similar to the distribution function W(r|open) of the short segment of persistence length .

Using Eqs. 13–15 and C5, we construct the effective initiation-termination distance distribution

(16)

W_eff(r) is plotted in Appendix C (Fig. 11) for various U₀. Qualitatively similar loop probability distributions have also been computed within the WLC model but without finite-sized molecules at the ends (Liverpool and Edwards, 1995). Here and in all subsequent analyses, we use the typical parameters /a = 0.2, d = a, and /a = 0.1. As U₀ is increased, the distance distribution function switches over from W(r|open) to W(r|loop). The statistics of W(r|open) and W(r|loop) are governed by L_N = Na and L_mn = (m + n), respectively. The loop forming factors, since they are close to the initiation and termination sites (L_mn << L_N), enhance the probability that the ends are close to each other, particularly when the binding energy U₀ is large.

View larger version (29K): [in this window] [in a new window]   FIGURE 11  (A) FJC and WLC models for W(r|open) for /a = 0.2. The WLC distribution approximates that of the FJC if the effective number of persistence lengths N is reduced. This reduction compensates for the stiffness of the chain that tends to give more weight at larger distances. (B) FJC and WLC distributions for /a = 1. Note the heuristic cutoff applied to the WLC model at r = a. As expected, for equal N, the WLC model gives a typically larger separation and hence smaller a/R; however, a/RN-1/2 for N in all cases. (C) The effective end-to-end distance distribution Weff constructed from W(r|open) via Eqs. 14 and 15.

(17)

View larger version (21K): [in this window] [in a new window]   FIGURE 5  The effective diffusional distance or harmonic distance over which recycled ribosomes must diffuse. (A) The dependence of R/a as a function of loop binding energy U0 is shown for N = 100 persistence lengths of coding mRNA. For large binding energies U0, the initiation and termination sites are brought closer together. The crossover between the end-to-end distribution function of a free chain to that of a loop occurs near Increasing the length of the short noncoding ends of the mRNA predominantly increases the typical distance R in the large U0, looped regime. (B) The N-dependence of R/a with the ratio of noncoding persistence lengths to coding persistence lengths (m + n)/N = 1/2. The N-dependence manifests itself primarily in the low U0, open chain regime. (C) The N dependence of R/a for various U0.

We now couple our mathematical models by incorporating the W_eff-weighted inverse harmonic distance a/R into the local, effective concentration C(a;R) given by Eq. 10. The effective injection rates = kC(a) that control the translation rate within the steady-state TASEP are then self-consistently determined.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION PHYSICAL MODELS RESULTS AND DISCUSSION EXPERIMENTAL CONSEQUENCES AND... SUMMARY APPENDIX A: PHYSICAL ASSUMPTIONS... APPENDIX B: MEAN FIELD... APPENDIX C: OPEN CHAIN... APPENDIX D: ASYMPTOTICS FOR... ACKNOWLEDGEMENTS REFERENCES

 
Here, we compute the possible currents J and the parameter space in which each are valid. We will use the exact solution Eq. 1, or its three asymptotic forms (Eq. 3), as well as J = kC(a)(1 - ₁) in Eq. 10, to find all relevant quantities and parameter phase boundaries.

Substitution of J = kC(a)(1 - ₁) into Eq. 10 and solving for ₁, we find

(18)

Upon multiplying Eq. 18 by kC(a), we find

(19)

To find C(a) in terms of known parameters, we use the explicit solutions of the TASEP for the current J(kC(a), ß, p) (Eq. 1 or 3) as indicated on the right-hand side of Eq. 19. The exact solution Eq. 1 yields an N + 2 order equation in kC(a) which we solve numerically. Only one of the N + 2 roots of Eq. 19 is real, yields occupations between zero and one, and is the physically relevant. The self-consistent solutions for kC(a) are used to evaluate J(kC(a), ß, p), which are plotted in Fig. 6, A and B. As expected, shorter chains yield slightly higher current. Larger D also increases the current and makes the approximate maximal current phase obtainable at smaller kC/p. Asymptotic limits for the current near phase boundaries and at large N are given in Appendix D.

View larger version (21K): [in this window] [in a new window]   FIGURE 6  The numerically determined, steady-state currents at finite N. The self-consistent currents were found by numerically finding the roots to the polynomial in J obtained by substituting the last line of Eq. 19 into the exact Eq. 1. (A) Steady-state currents as a function of the injection rate kC/p for R/a = 3 and various for N = 10, and N = 50 are compared. (B) J as a function of length N for and kC/p = 0.3,1. The current is relatively insensitive to N for

First assume that the detachment rate ß p/2 and consider the maximal current (phase III in the TASEP) where J = p/4. This occurs when both , ß > p/2. To determine the parameter regime in which J = p/4 holds, we solve for C(a) and determine for what range of parameters = kC(a) > p/2. Using J = p/4 in Eq. 19, we find

(20)

The criterion for maximal current, k > p/(2C(a)), is thus

(21)

Upon solving Eq. 21 for k, we find the minimum k = k^* required to achieve maximal current J = p/4:

(22)

Note that for large enough p/(4aDC) the critical value k^* can diverge. The divergence is more likely for larger R and occurs when there is simply not enough ribosome nearby to provide a large enough "on" rate to achieve maximal current. Even when the source (termination end) is held at the initiation site (R = a), there is the possibility that k^*, and maximal current, are never attained. This behavior arises because even for ribosomes released at an infinitely absorbing spherical initiation surface, there is a probability of escape (Berg and Purcell, 1977).

Next, let us consider small ß and large = kC(a). The mRNA has a high ribosome occupancy and a steady-state current J = ß(1 - ß/p). This regime (phase II) is termination rate-limited and occurs for ß < p/2 and ß < = kC(a). Upon using J = ß(1 - ß/p) in Eq. 19,

(23)

The only physical range of ß that satisfies Eq. 23 is

(24)

where Equation 24 defines the phase boundary between the high-density, exit rate-limited phase II and the low-density, initiation rate-limited phase I. This phase boundary is plotted as a function of kC/p for fixed 4aDC/p = 0.5 in Fig. 7 B. In the limit the phase boundary straightens as in the standard TASEP and is approximately

(25)

View larger version (23K): [in this window] [in a new window]   FIGURE 7  The modified phase diagram for translation rates along long (N ) mRNAs. (A) The minimum binding rate (Eq. 22) required to support the maximal current phase assuming that ß > 1/2. This value depends on the bulk ribosome concentration C and the distance R between the initiation and termination sites. (B) The modified phase diagrams as functions of kC/p for 4aDC/p = 0.5 and various R/a. (C) Modified phase diagrams as functions of kC/p for fixed and R/a = 10.

(26)

In the limit

(27)

which reduces to the result one would expect from infinitely fast initiation site clearance.

Summarizing, the large N, steady-state ribosome currents (given by Eq. 1), in terms of ribosome concentrations and kinetic "on" rates, are

(28)

where kC(a) in phase I is expressed in terms of known parameters according to Eq. 19. The mean occupations of the initiation and termination sites, in each regime, can now be readily found. At the first site, ₁ = 1 - J/(kC(a)), where we use J = J_L, J_R, or J_max (currents associated with each phase), and kC(a) found from Eqs. 19, 23, or 20. Similarly, the occupation at the last site is _N = J/ß. All of our results can be expressed in terms of three of the four nondimensional parameters: kC/p, 4aDC/p, and R/a. We shall present our results in terms of the relevant nondimensional parameters appropriate for the discussion at hand. For example, if the binding rate k is controlled as an independent variable, we use kC/p, 4aDC/p, and R/a as the governing parameters. If the bulk concentration and the diffusion constant are experimentally tuned, then our results should be expressed in terms of kC/p, and R/a.

Fig. 7 A shows the critical value k^*, above which an N TASEP is in the maximal current phase (provided ß/p > 1/2). When C is small and p is large, there is not enough ribosome in the cytoplasm to feed the initiation fast enough compared to the clearance rate p. Therefore the maximal current (J = p/4) arises only when the binding is efficient and k > k^* is large. For smaller R (termination site close to the initiation site), smaller values of 4aDC/p can still support maximal current. From Eq. 22, we see that when 4aDC/p (1 - a/(2R))/2, the critical value k^* diverges and the maximal current can never be reached. There is simply not enough ribosomes or the diffusion is too slow for there to be sufficient concentration at the initiation site to support the maximal current phase.

If the diffusion constants D and C are chosen such that, for example, 4aDC/p is small, the critical values k^* vary considerably with R/a, as shown by the points (4aDC/p = 1/2) in Fig. 7 A. The effects of depletion arise suddenly, with onset only at values of For large R/a, values of 4aDC/p 0.5 will render the critical k^* values very sensitive to R. If the initiation site has an interaction size of a 10 nm, and p 2–3/s (20–30 codons/s; Kruger et al., 1998), a diffusion constant of D 10^-8 cm²/s requires an effective concentration of C 0.01 - 0.02 µM for the phase diagram to be sensitive to diffusional depletion and R. Although typical total cytoplasmic ribosome concentrations are C 1 µM, many components must assemble to activate a translation-viable ribosome. For example, eIF4F exists at 0.01–0.2x the total ribosome concentration (Duncan et al., 1987). Furthermore, this already low abundance of eIF often needs to be further phosphorylated to be active. Thus, the effective concentrations C (and even diffusion constants) appropriate for our model may very well be low enough to fall within the range for the phase boundaries to be extremely sensitive to diffusional effects.

Fig. 7, B and C, show the steady-state phase diagrams as functions of ß/p and effective binding rate kC/p. In these phase diagrams, as in the unperturbed ones defined by Eq. 3, the upper-left region corresponds to a low density phase, the lower-right region corresponds to a high density phase, and the upper-right region describes a half-occupied (except near the ends i = 1,N), maximal current phase. The current J is constant throughout the maximal current phase and is not changed if kC/p or ß is increased beyond k^*C/p and 1/2, respectively. The phase diagram is modified by ribosome diffusion and depletion near the initiation site. The unmodified phase boundary between phases I and II of the TASEP (Eq. 3) would simply be defined by the straight line segment ß/p = kC/p. The main effects of diffusional depletion (by the initiation sink) and replenishment (by the termination source) on the standard phase diagram Fig. 3 is to shift the low density-maximal current phase boundary to larger effective injection rates kC/p and bend the low density-high density phase boundaries accordingly. Fig. 7 B depicts the phase boundaries defined by Eqs. 22 and 24 for fixed R/a = 3/2,4,10, , and fixed 4aDC/p = 1/2 as indicated by the points in Fig. 7 A. In this example k^*C/p = 3/2,4,10 for R/a = 3/2,4,10, respectively. Note that for R/a that k^* diverges and the maximal current phase is never attained. If 4aDC/p < 1/2, then there will be a finite value of R/a such that k^* diverges.

If, instead, is held fixed, the phase boundaries are nearly straight, as shown in Fig. 7 C. Here, we fixed R/a = 10, and plotted the phase diagrams for The corresponding values of kC/p above which the maximal current phase is attained are and 79/120, respectively.

Our results up to this point are contingent on the fact that measurements are averaged over timescales such that the TASEP and the diffusion processes have reached steady state, and the mRNA chain distribution has thermally equilibrated. The possibility exists that the chain conformations are not in thermodynamic equilibrium while the TASEP and the bulk ribosome diffusion has reached steady state for a given chain conformation. Thus, although not relevant within each of the three well-defined physical processes, the issue of kinetic versus thermodynamic control of ribosome throughput arises when one considers measurements over timescales that are insufficient to allow equilibration of the mRNA chain. The consequences of this are discussed in the following section.

	EXPERIMENTAL CONSEQUENCES AND PROPOSED MEASUREMENTS

TOP ABSTRACT INTRODUCTION PHYSICAL MODELS RESULTS AND DISCUSSION EXPERIMENTAL CONSEQUENCES AND... SUMMARY APPENDIX A: PHYSICAL ASSUMPTIONS... APPENDIX B: MEAN FIELD... APPENDIX C: OPEN CHAIN... APPENDIX D: ASYMPTOTICS FOR... ACKNOWLEDGEMENTS REFERENCES

 
The basic physical mechanisms described in our model for mRNA translation suggest a number of experimental tests. However, it must be emphasized that the model is meant to provide qualitative guidelines most useful for studying trends and how they depend on physical parameters. Translation occurring in vivo involve an enormous number of molecular species and biochemical processes to be quantitatively modeled, especially in the absence of significantly more detailed experimental findings. Nonetheless, our proposed mechanisms can be probed with carefully designed, simplified, in vitro experiments. Here, we discuss in detail the basic expected phenomena and their regimes of validity.

First note from Fig. 6, and from Appendix D, that the exact currents for a finite number of codons N very rapidly approach the asymptotic values given by Eq. 3 as N increases. Even when N is only 10–50, the steady-state ribosome currents are only a few percent off the exact N = results. In other words, the exact solution Eq. 1 is a very good approximation to Eq. 3 for Therefore, as a mental guide, it is typically sufficient to consider the currents J corresponding to an infinite chain (N = ) given by Eq. 3, but nonetheless consider a finite initiation-termination separation (measured by the harmonic distance R).

Polysomal density variations
Although we have focused on the steady-state current, the particle (ribosome) densities in each of the three current regimes are different and may be detected. In the TASEP model, the ribosome density profiles along the mRNA chain vary only near the initiation and termination ends. In the interior of the mRNA, the density is relatively uniform and are given by the last column in Eq. 3. In the exit-rate limited phase (small ß/p), where J = ß(1 - ß/p), the midpoint density _N/2 1 - ß/p is high, whereas in the low injection rate case, J = (1 - /p), and _N/2 /p is low. The typical density in the maximal current regime is 1/2. These densities are also approximately correct when one explicitly treats large ribosomes that occlude many codon "lattice sites." Therefore, we might expect that one may be able to predict in which current regime translating mRNA exists if ribosome densities can be estimated from images taken with, e.g., AFM or electron micrograph techniques. For example, in Fig. 1 A, the high density of ribosomes suggests that the system is in phase II whereas the steady-state current J = ß(1 - ß/p) is a function only of the detachment rate ß.

Kinetic binding rate and ribosome concentration dependences
Fig. 7 A shows the minimum effective attachment rate k^*C/p necessary for a large system to be in the maximal current regime (where the ribosome current J p/4) as a function of the effective ribosome diffusion constant. An additional requirement is that the effective detachment rate ß/p > 1/2. The value of k^* can be tuned perhaps by substitution of the codons comprising the initiation sites, or by other physical means. Although ribosome diffusion constants are difficult to vary over a wide range (by modifying the solution viscosity), the critical k^* is a very sensitive function of D, particularly for small D. It is thus possible that slightly increasing the ribosome diffusivity can dramatically decrease the k^* necessary for the system to be in the maximal current regime.

As mentioned, changing the mRNA length N does not significantly affect the overall steady-state current along the chain (beyond about N 10–20) but it can change the statistics of the initiation-termination separation by changing R. Increasing the harmonic separation R has qualitatively the same effect as decreasing the ribosome diffusivity, since terminated ribosomes now have further to diffuse back to the initiation site. For

(29)

the maximal current regime is never reached. This can be easily seen from Eq. 22. Thus, rather than tuning the ribosome diffusivity, decreasing C may preclude the system from entering the maximal current phase if Eq. 29 is satisfied. There is simply not enough ribosome available for sufficient initiation to be achieved so that the maximal current phase arises.

When Eq. 29 is not satisfied, the maximal current phase can exist. In Fig. 8 A, we replot the phase diagram corresponding to R/a = 10 shown in Fig. 7 C. Fixing the parameter 4aDC/p = 0.6 allows k to be the only free parameter. This kinetic "on" rate k can be tuned by varying ribosome recruitment proteins such as eIF4E. If ß/p > 1/2, C, D, and p are held constant, increasing k from a sufficiently small value allows one to traverse the trajectory S₁. The steady-state ribosome current starts in the low density phase I with current given by Eq. 26. As k is increased, the steady-state current increases until it continuously crosses over into the maximal current regime (phase III), where the ribosome throughput is given by J = p/4. Further increasing k when inside the maximal current phase III will no longer affect the steady-state ribosome current. If, however, ß/p < 1/2, the current behavior abruptly crosses over (along trajectory S₂) from that given by Eq. 26 to J = ß(1 - ß/p) corresponding to the high ribosome density phase II. In this phase the detachment step is rate-limiting, and further increases in k will no longer affect the throughput.

View larger version (19K): [in this window] [in a new window]   FIGURE 8  Large N phase diagrams for R/a = 10. (A) Phase diagram for fixed 4aDC/p = 0.6 with trajectories S1,2 corresponding to increasing kinetic "on" rate k. (B) Phase diagram when is fixed, and trajectories S3,4 correspond to increasing bulk ribosome concentration C. Trajectory S3 traverses the I–III phase boundaries for (thick curves) but not for (thin curves). Trajectory S4, on the other hand, traverses the I–II phase boundaries for both

Despite the apparent fundamental importance of the kinetic binding, or "on" rate in translation, there are no systematic and independent measurements of k in the literature. The required independent estimates of k may be achieved by perhaps combined kinetic and affinity measurements of the association of a minimal set of components, including only the ribosomes and a portion of the 5' initiation codons and co-factors. For the "off" rate ß, similar ideas can be employed. The tRNA or ribosome release