Article Information

• PDF (645 kb)
• Full Text with Large Figures

PubMed

• Articles by Hermioni Zouridis
• Articles by Vassily Hatzimanikatis

Related Articles

• Analysis of Binding at a Single Spatially Localized Cluster ...

  Analysis of Binding at a Single Spatially Localized Cluster of Binding Sites by Fluorescence Recovery after Photobleaching Biophysical Journal, Volume 91, Issue 4, 15 August 2006, Pages 1169-1191 Brian L. Sprague, Florian Müller, Robert L. Pego, Peter M. Bungay, Diana A. Stavreva and James G. McNally Abstract Cells contain many subcellular structures in which specialized proteins locally cluster. Binding interactions within such clusters may be analyzed in live cells using models for fluorescence recovery after photobleaching (FRAP). Here we analyze a three-dimensional FRAP model that accounts for a single spatially localized cluster of binding sites in the presence of both diffusion and impermeable boundaries. We demonstrate that models completely ignoring the spatial localization of binding yield poor estimates for the binding parameters within the binding site cluster. In contrast, we find that ignoring only the restricted axial height of the binding-site cluster is far less detrimental, thereby enabling the use of computationally less expensive models. We also identify simplified solutions to the FRAP model for limiting behaviors where either diffusion or binding dominate. We show how ignoring a role for diffusion can sometimes produce serious errors in binding parameter estimation. We illustrate application of the method by analyzing binding of a transcription factor, the glucocorticoid receptor, to a tandem array of mouse mammary tumor virus promoter sites in live cells, obtaining an estimate for an in vivo binding constant (10M), and a first approximation of an upper bound on the transcription-factor residence time at the promoter (∼170ms). These FRAP analysis tools will be important for measuring key cellular binding parameters necessary for a complete and accurate description of the networks that regulate cellular behavior. Abstract | Full Text | PDF (635 kb)

• Analysis of Binding Reactions by Fluorescence Recovery after...

  Analysis of Binding Reactions by Fluorescence Recovery after Photobleaching Biophysical Journal, Volume 86, Issue 6, 1 June 2004, Pages 3473-3495 Brian L. Sprague, Robert L. Pego, Diana A. Stavreva and James G. McNally Abstract Fluorescence recovery after photobleaching (FRAP) is now widely used to investigate binding interactions in live cells. Although various idealized solutions have been identified for the reaction-diffusion equations that govern FRAP, there has been no comprehensive analysis or systematic approach to serve as a guide for extracting binding information from an arbitrary FRAP curve. Here we present a complete solution to the FRAP reaction-diffusion equations for either single or multiple independent binding interactions, and then relate our solution to the various idealized cases. This yields a coherent approach to extract binding information from FRAP data which we have applied to the question of transcription factor mobility in the nucleus. We show that within the nucleus, the glucocorticoid receptor is transiently bound to a single state, with each molecule binding on average 65 sites per second. This rapid sampling is likely to be important in finding a specific promoter target sequence. Further we show that this predominant binding state is not the nuclear matrix, as some studies have suggested. We illustrate how our analysis provides several self-consistency checks on a FRAP fit. We also define constraints on what can be estimated from FRAP data, show that diffusion should play a key role in many FRAP recoveries, and provide tools to test its contribution. Overall our approach establishes a more general framework to assess the role of diffusion, the number of binding states, and the binding constants underlying a FRAP recovery. Abstract | Full Text | PDF (948 kb)

• Fluctuations and Slow Variables in Genetic Networks

  Fluctuations and Slow Variables in Genetic Networks Biophysical Journal, Volume 84, Issue 3, 1 March 2003, Pages 1606-1615 R. Bundschuh, F. Hayot and C. Jayaprakash Abstract Computer simulations of large genetic networks are often extremely time consuming because, in addition to the biologically interesting translation and transcription reactions, many less interesting reactions like DNA binding and dimerizations have to be simulated. It is desirable to use the fact that the latter occur on much faster timescales than the former to eliminate the fast and uninteresting reactions and to obtain effective models of the slow reactions only. We use three examples of self-regulatory networks to show that the usual reduction methods where one obtains a system of equations of the Hill type fail to capture the fluctuations that these networks exhibit due to the small number of molecules; moreover, they may even miss describing the behavior of the average number of proteins. We identify the inclusion of fast-varying variables in the effective description as the cause for the failure of the traditional schemes. We suggest a different effective description, which entails the introduction of an additional species, not present in the original networks, that is slowly varying. We show that this description allows for a very efficient simulation of the reduced system while retaining the correct fluctuations and behavior of the full system. This approach ought to be applicable to a wide range of genetic networks. Abstract | Full Text | PDF (145 kb)

• …more

Copyright © 2008 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 95, Issue 3, 1018-1033, 1 August 2008

doi:10.1529/biophysj.107.126128

Biophysical Theory and Modeling

Effects of Codon Distributions and tRNA Competition on Protein Translation

Hermioni Zouridis^* and Vassily Hatzimanikatis^†, ,  

^* Department of Chemical and Biological Engineering, McCormick School of Engineering and Applied Sciences, Northwestern University, Evanston, Illinois
^† Laboratory of Computational Systems Biotechnology, École Polytechnique Fédérale de Lausanne, CH-1015, Lausanne, Switzerland

Address reprint requests to Vassily Hatzimanikatis, Tel.: 41-0-21-693-98-70.

Translation, or protein synthesis, is a process that is central to cellular function. It is essentially a template polymerization process 1 consisting of initiation, elongation, and termination phases. Messenger RNA (mRNA), composed of a sequence of codons coding for amino acids, carries genetic information. Initiation occurs with binding of the ribosome to the ribosomal binding site near the 5′ end of the mRNA. During the elongation phase the ribosome facilitates assembly of the polypeptide chain with one amino acid (aa) added per elongation cycle at each codon. Amino acids are delivered to the ribosome by transfer RNAs (tRNAs) in the form of ternary complexes that serve as adaptor molecules between the amino acid and the codon present in the ribosomal A site. Termination involves release of the completed peptide from the ribosome near the 3′ end of the mRNA. Multiple proteins can be synthesized simultaneously on a single mRNA molecule, forming a structure called the polysome (or polyribosome) consisting of several ribosomes simultaneously translating the same mRNA. Polysome size is the number of ribosomes bound to a single mRNA molecule. Hence, the higher the polysome size, the greater the coverage of the mRNA due to ribosomes translating it. Polysomes have been observed experimentally 2, and modern techniques have allowed the quantification of polysome size for almost every mRNA in yeast cells 3.

The sheer complexity of the translation mechanism necessitates mathematical, mechanistic frameworks to better understand the system properties of translation and make quantitative predictions. Several studies have been conducted involving investigating the kinetics of protein synthesis that take into account the ribosome movement on mRNAs 4,5,6, and other studies 7,8 have involved the effects of competition for ribosomes between mRNAs on cell-wide mapping between mRNA and protein levels. An assumption in these studies is that the elongation kinetics at each codon depends on a single rate constant that is the same for all codon species at all positions along the length of the mRNA. In reality, codons have varying elongation kinetics due to different tRNA availabilities 9 and codon-anticodon compatibilities 10,11,12, and the multiple elementary steps and translational components involved in the elongation cycle at every codon. Therefore, a better understanding of the properties of translation requires the consideration of the translation elongation phase, accounting for all elongation cycle intermediate steps. In previous work 13, we developed such a kinetic model of the translational machinery that is deterministic and sequence-specific, and accounts for all the elementary steps of the translation mechanism. We also performed a sensitivity analysis to determine the effects of the kinetic parameters and concentrations of the translational components on the protein synthesis rate.

A finding from our model was that tRNA concentrations have almost no impact on protein synthesis rate. However, experimental evidence suggests tRNA concentrations are significant to translation kinetics. The work by Ikemura 14 shows a correlation between tRNA abundances and codon frequencies. Other work demonstrates that synonymous codons (different codons coding for the same amino acid) are not translated at the same rate 11, with higher translation rates for more abundant or major codons 15. Given the difference between experimental results and those determined from our computational studies 13, it is important to note that a simplifying assumption made in our model is that only ternary complexes that recognize the A site codon can bind to the ribosome. In reality, ternary complexes initially bind nonspecifically to the ribosomal A site, which means that both ternary complexes recognizing and not recognizing the ribosomal A site codon can bind to the ribosome in the first intermediate step of the elongation cycle of each codon. The experimentally observed importance of tRNA concentration to protein synthesis kinetics, coupled with our observation that tRNA concentrations are not scarce enough to modulate translation rate, motivates questions about the role the competition between ternary complexes for ribosomal A site binding plays in protein synthesis kinetics.

Hence, in this work we expand our mechanistic framework to account for ternary complex competitive binding to the ribosomal A site. We also expand our sensitivity analysis to make it codon-specific, meaning that we account for the contribution of kinetic parameters and translational component concentrations of each codon on the overall protein synthesis rate. We find that our expanded mechanistic framework predicts lower protein synthesis rates than our previous framework 13. Our sensitivity analysis predicts that, at low polysome sizes, the codons near the 5′ end of the mRNA control protein synthesis rate, at intermediate polysome sizes different configurations of codons along the length of the mRNA control protein synthesis rate, and at high polysome sizes the codons near the 3′ end of the mRNA control protein synthesis rate. Moreover, our sensitivity analysis identifies the competitive, nonspecific binding of the tRNAs to the ribosomal A site as rate-limiting to the elongation cycle for every codon. By introducing our previous 13 and current mechanistic models in terms of the Michaelis-Menten kinetic framework, we determine that these results are due to the tRNAs that do not recognize the ribosomal A site codon acting as competitive inhibitors to the tRNAs that do recognize the ribosomal A site codon. We also observe that the relative position of codons along the mRNA determines the optimal protein synthesis rate, and that the translation rates of mRNAs are controlled by segments of rate-limiting codons that are sequence-specific.

The translation elongation phase is a cyclic process that involves codons, ribosomes, amino acids, tRNAs, elongation factors Tu, Ts, and G, and leads to the assembly of polypeptide chains (Fig. 1). Each amino acyl-tRNA (aa-tRNA) binds to Ef-Tu:GTP, forming a ternary complex (step 13). The ternary complex then binds reversibly to the ribosomal A site in a codon-independent manner (step 1). After finding the correct codon match and reversible codon-dependent binding (step 2), GTP is hydrolyzed (step 3), Ef-Tu:GDP changes position on the ribosome (step 4) and is released (step 5). In a two-step process, Ef-Ts catalyzes regeneration of Ef-Tu:GTP (steps 11 and 12). During accommodation the aa-tRNA undergoes a conformation change and enters the A site (step 6). Transpeptidation then occurs (step 7), where the peptide chain is transferred from the peptidyl-tRNA to the aa-tRNA, resulting in the elongation of the polypeptide chain by one amino acid. Reversible binding of Ef-G:GTP (step 8) facilitates translocation (step 9). During translocation the P site tRNA and codon move to the E site of the ribosome and the A site tRNA and codon move to the P site, resulting in the complex moving toward the 3′ end of the mRNA by one codon. The tRNA in the E site is released along with Ef-G:GDP (step 10), and Ef-G:GTP is recycled in a two-step process (steps 14 and 15).

Display large version of this figure

Figure 1

The elementary mechanistic steps of the translation elongation process. Ribosomal A, P, and E sites indicated on the intermediates between steps 1 and 2 and steps 9 and 10. Step 1: Reversible, codon-independent binding of the ternary complex to the ribosomal A site. Step 2: Reversible, codon-dependent binding of the ternary complex to the ribosomal A site. Step 3: GTP hydrolysis. Step 4: Ef-Tu:GDP position change on the ribosome. Step 5: Ef-Tu:GDP release. Step 6: aa-tRNA accommodation. Step 7: Transpeptidation. Step 8: Reversible binding of Ef-G:GTP. Step 9: Translocation. Step 10: E site tRNA release. Steps 11 and 12: Ef-Ts catalyzed regeneration of Ef-Tu:GTP. Step 13: Ef-Tu:GTP binding to the aa-tRNA. Steps 14 and 15: Regeneration of Ef-G:GTP.

In this section, we introduce a mechanistic framework that incorporates the kinetics of all the intermediate steps of the translation elongation cycle occurring at a given codon in a single expression. A summary of the assumptions made in this formulation, along with descriptions of the variables and parameters, can be found in Appendix A . A detailed description of this model can be found in our previous study 13.

The initiation rate is described as

(1)

The elongation rate at codon n along the length of the mRNA species r is described as

(2)

The effective elongation rate constant at codon position n, (Eq. (3)), is comprised of terms representing the kinetics of each of the translation elongation cycle intermediate steps occurring at that codon, and these terms depend on the reaction rate constants corresponding to the elongation cycle intermediate steps 13:

(3)

(4)

(5)

(6)

(7)

Table 1 Effective elongation rate constant terms

	Parameter	Expression	Elongation cycle intermediate step	Magnitude
	α1,j		Codon-independent binding of the ternary complex, noncompetitive conditions.	6×10−4−0.04
			Codon-independent binding of the ternary complex, competitive conditions.	0.19–12.9
	α2		Codon-dependent binding.	0.005
	α3	1/k3	GTP hydrolysis.	0.01
	α4	1/k4	Ef-G:GDP position change on ribosome.	0.0015
	α5	1/k5	Ef-G:GDP release.	0.067
	α6	1/k6	A site tRNA accommodation.	0.05
	α7		Ef-G:GTP binding.	3.5×10−4
	α8	1/k8	Translocation.	0.004
	α9	1/k9	E site tRNA release.	0.05

Numerical values for the reaction rate constants, k1, k−1, k2, k−2, k3, k4, k5, k6, k7, k−7, k8, and k9, the Ef-G concentration, G(f), are included in Zouridis and Hatzimanikatis 13. Experimental data for the reaction rate constants can be found in the literature 24,25,26,27. Experimental data for the Ef-G concentration can be found in Hershey 1. Numerical values for the free ternary complex species concentrations, , are included in Table 2.

The dynamics describing the transition between the states of the elongation phase are as follows:

(8)

(9)

(10)

The total ribosome and codon concentrations are expressed by Eqs. (11), respectively,

(11)

(12)

We investigate the effects of elongation cycle kinetics at each codon along the length of the mRNA on the steady-state protein synthesis rate by examining the flux control coefficients, which are defined as fractional flux changes with respect to fractional input parameter changes 16. Similar to the Summation Theorem 16, we can show that the sum of the control coefficients with respect to the reaction rate constants for an mRNA species that is not competing for translational resources with other mRNA species is equal to one,

(13)

(14)

(15)

We utilize our mathematical model of protein synthesis and the sensitivity analysis to investigate the steady-state translation properties of Escherichia coli mRNAs as functions of polysome size with and without accounting for ternary complex competitive binding to the ribosomal A site. Polysome size is the number of ribosomes bound to a single mRNA molecule, so the higher the polysome size, the greater the coverage of the mRNA by ribosomes. Hence, we define ρ to be the fraction of the mRNA molecule covered by translating ribosomes, as

(16)

In this work, we consider three cases with respect to ternary complex binding to the ribosomal A site codon. These cases differ by how the free ternary complex concentrations are applied for the quantification of the effective elongation rate constants and . In Table 2 we list the free ternary complex concentrations and corresponding magnitudes for and for all the ternary complex species. The effective elongation rate constant magnitudes shown correspond to U_n,r=1. The following assumptions were employed for each case:

Table 2 Effective elongation rate constant magnitudes for each E. coli ternary complex species

	Species	T(f) (μM)	*	*	Species	T(f) (μM)	*	*
	Ala1B	14.7	5.3	2.0	Leu5	3.5	5.1	0.7
	Ala2	1.9	4.9	0.4	Lys	6.1	5.2	1.1
	Arg2	23.1	5.3	2.7	Met	2.5	5.0	0.5
	Arg3	2.3	5.0	0.4	Phe	4.0	5.1	0.7
	Arg4	3.4	5.1	0.6	Pro1	2.2	5.0	0.4
	Arg5	2.3	5.0	0.4	Pro2	3.8	5.1	0.7
	Asn	5.4	5.2	0.9	Pro3	2.0	4.9	0.4
	Asp1	9.5	5.2	1.5	Ser1	6.9	5.2	1.2
	Cys	7.2	5.2	1.2	Ser2	1.3	4.7	0.3
	Gln1	3.0	5.0	0.6	Ser3	5.4	5.2	1.0
	Gln2	3.8	5.1	0.7	Ser5	3.8	5.1	0.7
	Glu2	20.7	5.3	2.5	Thr1	0.4	3.7	0.1
	Gly1	5.6	5.2	1.0	Thr2	2.8	5.0	0.5
	Gly2	5.6	5.2	1.0	Thr3	4.3	5.1	0.8
	Gly3	18.0	5.3	2.3	Thr4	5.4	5.2	1.0
	His	3.6	5.1	0.7	Trp	4.1	5.1	0.7
	Ile1	7.2	5.2	1.2	Tyr1	4.4	5.1	0.8
	Ile2	10.8	5.3	1.6	Tyr2	5.1	5.2	0.9
	Leu1	14.9	5.3	2.0	Val1	17.1	5.3	2.2
	Leu2	5.2	5.2	0.9	Val2A	2.6	5.0	0.5
	Leu3	2.2	5.0	0.4	Val2B	3.7	5.1	0.69
	Leu4	9.4	5.2	1.5

Numerical values for the free ternary complex species concentrations, T(f), are estimated in Zouridis and Hatzimanikatis 13. Experimental data for the ternary complex concentrations can be found in Dong et al. 23.

* Evaluated at Un,r=1.

Only the ternary complex species recognizing the ribosomal A site codon are allowed to participate in the nonspecific binding step of the elongation cycle. In studies considering Case I, the ternary complex concentrations recognizing the A site codons, , are set equal to the median concentration, , in the effective elongation rate constant expressions of every codon. Although variations in ternary complex concentrations cause variations in effective elongation rate constant magnitudes, we have observed that these differences are negligibly small under noncompetitive binding conditions 13.

All ternary complex species are allowed to participate in the nonspecific binding step of the elongation cycle. Similar to Case I, in studies considering Case II, the ternary complex concentrations recognizing the A site codons, , are set equal to the median concentration, in the effective elongation rate constant expressions of every codon. Although we observe in this work that variations in ternary complex concentrations cause significant variations in elongation rate-constant magnitudes, Case II allows us to study the effects of ternary complex competitive binding in a codon-independent manner. Moreover, because all codons are treated uniformly in Case II, we study the effects of ternary complex competitive binding in a sequence-independent manner.

Similar to Case II, all ternary complex species are allowed to participate in the nonspecific binding step of the elongation cycle. However, in studies considering Case III, the ternary complex concentrations recognizing the A site codons, are set equal to their respective physiological levels in the effective elongation rate constant expression Because codons are not treated uniformly in Case III, we study the effects of ternary complex competitive binding in both a codon- and sequence-specific manner.

In these studies we apply Cases I and II to investigate the translation properties of the trpR gene of E. coli in both a codon- and sequence-independent manner.

We observe that as ribosomal fractional coverage increases, the protein synthesis rate increases, reaches a maximum, and then decreases under both competitive (Fig. 2, curves ii and iii) and noncompetitive (Fig. 2, curve i) binding conditions. Included in Fig. 2 are results for Cases I and II (curves i and iii, respectively), and Case II with all the codons in the sequence recognized by the ternary complex species having the maximum free concentration of 23.1μM (curve ii). The translation rates determined under Case I are higher at each polysome size than those determined under Case II. This result is due to the large difference in the effective elongation rate constant magnitudes under the two cases. For U_n,r=1, under Case I, (curve i), while under Case II, (curve ii) and (curve iii). Because the effective elongation rate constant magnitudes under Case I are higher than those under Case II, the translation rates observed under Case I are higher than those observed under Case II.

Display large version of this figure

Figure 2

Relationship between translation rate and polysome size for (i) Case I, and Case II with all codons recognized by the ternary complex species of (ii) maximum and (iii) median concentrations.

We applied the control analysis framework to the model to determine if translation is initiation-, elongation-, or termination-limited under different polysome sizes. We observe that under both Cases I and II, translation is initiation-limited for ρ<0.5; elongation-limited for 0.5<ρ<0.99, with elongation control maximal at the same ribosomal fractional coverage that specific protein production rate is maximal; and termination-limited for ρ>0.99.

We investigated how control of the elongation phase over translation rate is distributed with respect to the codons along the length of the mRNA at different polysome sizes by examining the control coefficients corresponding to the effective elongation rate constants, (Fig. 3). We observe that at low polysome sizes the elongation phase control over translation rate lies in the codons near the 5′ end of the mRNA. This result is in agreement with early experimental results demonstrating that point mutations near the start codon of the mRNA cause dramatic changes in protein expression levels 28,29. Also, at intermediate polysome sizes the control is distributed along the length of the mRNA in different configurations, and at high polysome sizes the control lies in the codons near the 3′ end of the mRNA. We observe the same results under both Cases I and II. These results are expected because at low polysome sizes kinetics are initiation-limited (see previous paragraph for discussion), which means that the initiation process limits the progress of protein translation. Hence, the more efficiently the codons near the 5′ end of the mRNA can be translated, the more ribosomes can be transferred to downstream codons along the length of the sequence. Faster transfer of ribosomes due to more efficient translation of these codons elevates protein synthesis rate by increasing the probability of an initiation event occurring without changes to the initiation process being made. The converse is true for termination-limited conditions.

Display large version of this figure

Figure 3

Elongation step control coefficients, , with respect to sequence position under initiation (A), elongation (B), and termination (C) limited conditions for Cases I and II.

We investigated how the elongation phase control is distributed with respect to the elongation cycle intermediate steps at each codon along the length of the mRNA by examining the control coefficients: , and , along with the control coefficients corresponding to free ternary complex concentration, . We observe that the rate-limiting step at each codon along the length of the mRNA is different between Cases I and II. Under Case I, we observe that the control coefficient with respect to the Ef-Tu:GDP release rate constant, , is the highest of the control coefficients corresponding to elongation cycle intermediate steps at every sequence position and polysome size (Figure 4A, results shown only for ρ=0.67), indicating that this intermediate step is rate-limiting to the elongation cycle. This result is consistent with experimental reports which identify Ef-Tu:GDP release as one of the rate-limiting steps of the elongation cycle at a given codon 26. Control coefficients for A site tRNA accommodation and E site tRNA release are equal to each other and also high (Figure 4A, results shown only for ρ=0.67) at every sequence position and polysome size. The remaining elongation cycle intermediate steps have low control coefficients, including that for the free ternary complex concentration control coefficient .

Display large version of this figure

Figure 4

Elongation cycle intermediate control coefficients with respect to sequence position under Case I (A) and Case II (B) binding conditions. Results shown are for elongation-limited conditions (ρ=0.67).

However, under Case II we observe that the control coefficient with respect to the free ternary complex concentration, , is highest at every sequence position and polysome size (Figure 4B, results shown only for ρ=0.67), indicating that ternary complex nonspecific binding is rate-limiting to the elongation cycle. The remaining elongation cycle control coefficients are close to zero, indicating that the intermediate steps after ternary complex nonspecific binding have very little influence on elongation cycle kinetics. Moreover, the rate-limiting effects of ternary complex nonspecific binding are much higher under Case II than the rate-limiting effects of Ef-Tu:GDP release under Case I, with free ternary complex concentration control coefficients under Case II, and more than twice as high as Ef-Tu:GDP release control coefficients under Case I. We also observe that the concentrations of ternary complexes that do not recognize the A site codon, (k≠j), have an inhibitory effect on translation kinetics because the corresponding control coefficients for the combined concentration of the incorrect ternary complexes, , are negative (Figure 4B, results shown only for ρ=0.67), meaning that an increase in this concentration would cause a decrease in translation rate.

It is important to note that it is the relative magnitudes of the terms in the effective elongation rate constant, that play a significant role in the distribution of control with respect to the elongation cycle intermediate steps at each codon. The influence each elongation cycle intermediate step has over the overall kinetics of the elongation cycle at a given codon is proportionate to the magnitude of its corresponding term in the effective elongation rate constant. Under noncompetitive binding conditions, Ef-Tu:GDP release is rate-limiting, with α₅=0.067 (Table 1) being the largest term in , and nonspecific ternary complex binding has almost no influence over elongation cycle kinetics, with α_1,j=6×10⁻⁴−0.04 (Table 1). However, under competitive binding conditions, the magnitude of the nonspecific ternary complex binding term in the effective elongation rate constant is much higher than α₅, with (Table 1), making nonspecific binding rate-limiting.

To further understand the relationship between the magnitudes of the effective elongation rate constant terms and the control the corresponding elongation cycle intermediate steps have over translation rate, we introduce the elasticities of the elongation rate at codon n with respect to the free ternary complex concentration, , and the reaction rate constant for Ef-Tu:GDP release, , under competitive binding conditions:

(17)

(18)

Elasticity is defined as the differential change in the rate of a single reaction step, i.e., in this case, V_ij,n,r. Unlike the control coefficients, which pertain to the overall translation rate of the mRNA, elasticity is therefore a property local to that reaction step and not a systemic property. However, due to the compactness of the elasticity expressions, they are useful for obtaining general quantitative insight into the impact of the individual reaction rate constants and translational components on their respective control coefficient magnitudes. It is evident from the above expressions that the elasticity of V_ij,n,r with respect to a given parameter is dependent on the effective elongation rate constant term to which the parameter pertains, and not only on that parameter. Along these lines, the relative magnitudes of the elasticities are proportionate to the relative magnitudes of the corresponding effective elongation rate constant terms, and similar relationships are obtained between the remaining effective elongation rate constant terms and their corresponding elasticities. Consequently, the control the elongation cycle intermediate steps have over the translation rate of each codon is strongly influenced by the magnitudes of their respective effective elongation rate constant terms.

Overall, in these studies we observe that ternary complex competitive binding to the ribosomal A site introduces changes to translation rate (Fig. 2). However, competitive binding does not cause changes to the distribution of overall initiation, elongation, and termination control with respect to polysome size. Moreover, competitive binding does not affect the codon-specific distribution of control with respect to polysome size (Fig. 3), but instead introduces changes to the distribution of control with respect to elongation cycle intermediate step at each codon (Fig. 4).

To further investigate the inhibitory effects of ternary complexes not recognizing the ribosomal A site codon, we derive our mechanistic framework in the context of Michaelis-Menten enzyme kinetics. Treating all the translating ribosomes in a single E. coli cell having codon species j occupying the A site as the enzyme, and the ternary complex species j recognizing the A site codon as the substrate, it can be shown that, in the absence of ternary complex competitive binding,

(19)

(20)

(21)

(22)

Under competitive binding conditions, the ternary complexes that do not recognize the A site codon (k≠j) bind to the ribosome as the ternary complexes do during the nonspecific binding step of the elongation cycle, but do not proceed to the subsequent intermediate steps. The ternary complexes (k≠j) occupying the ribosomal A site prevent the ternary complexes from binding to the ribosome, so the apparent affinity the ternary complexes have in recognizing the A site codon for the ribosome decreases. This decrease is due to the term multiplied by the Michaelis-Menten constant (K_M) in the expression for (Eq. (22)), which represents the inhibitory effects of the ternary complexes (k≠j) on translation rate. However, the maximum reaction rate is the same under both noncompetitive and competitive binding conditions. Fig. 5 shows the relationship between translation rates and as functions of the ternary complex concentration recognizing the A site codon, for the median ternary complex concentration, The maximum reaction rate is proportional to the concentration of translating ribosomes having the codon recognized by the ternary complex species of median concentration present in the A site, and the ternary complex concentration is allowed to vary. When evaluating the translation rate expressions with and without competitive binding at the median ternary complex concentration , we observe a much lower translation rate under competitive binding conditions than under noncompetitive binding conditions (Fig. 5). Similar results are observed for the remaining ternary complex species.

Display large version of this figure

Figure 5

Amino-acid rate of incorporation as a function of ternary complex concentration for Case I (solid line) and Case II (dashed line). Results shown are at the E. coli cellular level for the ternary complex species of median concentration. Similar results are observed for all ternary complex species.

We examined the expressions for the elasticities, and , of the reaction rates and with respect to the ternary complex concentration recognizing the ribosomal A site, i.e., the ratios of the proportional changes in and with respect to the proportional change in :

(23)

(24)

Evaluating Eqs. (23) yields

(25)

(26)

In these studies, we apply Case III to investigate the translation properties of mRNAs in both a codon and sequence-dependent manner. We applied our mechanistic framework to 100 randomly permuted sequences having identical codon frequencies representative of those of the E. coli genome. Each sequence is 361-codons-long, approximately the average length of an E. coli mRNA 22. Similar to our results in previous sections, we observe that the translation rate increases, reaches a maximum, and then decreases as polysome size increases (Fig. 6). However, optimum protein synthesis rates vary with sequence (Fig. 6, results shown only for sequences producing highest and lowest optimum translation rates). We also observe that the optimum rate occurs at multiple polysome sizes for each sequence and that there are regimes of polysome sizes for which translation properties are highly sensitive to the input parameters of our model (Fig. 6). Because all the sequences in this study have the same codon frequencies, the results presented in this section emphasize that the relative positions of codons along the length of the mRNA can influence protein synthesis properties.

Display large version of this figure

Figure 6

Relationship between translation rate and polysome size under Case III conditions.

To investigate the overall relationship between translation rate and polysome size with Case III conditions, we examined changes in the effective elongation rate constant magnitudes with polysome size. We scaled the effective elongation rate constants by dividing them by the effective elongation rate constant, , evaluated at U_n,r=1 and that has a magnitude of 0.8s⁻¹. In the absence of ribosomal crowding on the mRNA, i.e., when U_n,r=1, the scaled effective elongation rate constant magnitudes vary between 0.10 and 3.44 due to differences in the nonspecific binding term, (Eq. (5)). Using effective elongation rate constants determined with Case II conditions as a reference, we scaled them the same way by dividing them by the effective elongation rate constant, , evaluated at U_n,r=1 and that has a magnitude of 5.1s⁻¹. Included in Fig. 7 are the distributions of scaled effective elongation rate constant magnitudes as functions of sequence position under initiation (A), elongation (B), and termination (C) limited conditions for one of the sequences used in this section (similar results are observed for the other sequences). The dashed lines represent magnitudes under Case III conditions, and the solid lines represent magnitudes under Case II conditions.

Display large version of this figure

Figure 7

Scaled effective elongation rate constant magnitudes under initiation (A), elongation (B), and termination (C) limited conditions for one of the sequences used in this section (similar results are observed for the other sequences). The dashed lines represent magnitudes determined with Case III conditions, while the solid lines represent magnitudes determined with Case II conditions.

We observe that under initiation-limited conditions the scaled effective elongation rate constants for both Case II and Case III are approximately equal to the values they take on when U_n,r=1, and this result is expected because the polysome size is low and hence the mRNA is not crowded. Under elongation-limited conditions, the level of crowding on the mRNA is higher as reflected in the conditional probability term, U_n,r, of the effective elongation rate constant decreasing, which results in the scaled effective elongation rate constant magnitudes and translation rates increasing. The level of ribosomal crowding on the mRNA determines the magnitudes under Case II conditions (see 13 for more discussion), while the complex interplay between the level of ribosomal crowding on the mRNA and the level of ternary complex competition for the ribosomal A site at each codon determines the scaled effective elongation rate constant magnitudes under Case III conditions. Codons that experience a lot of ternary complex competitive inhibition have lower effective elongation rate constants and are hence translated more slowly than codons that do not, causing high ribosome density upstream on the mRNA and large variation in the conditional probability term, U_n,r, that is not observed under Case II conditions. Consequently the scaled effective elongation rate constants determined with Case III are much higher than those determined with Case II. Under termination limited conditions the polysome size is high, so crowding on the mRNA is maximal and U_n,r≈0, regardless of whether the binding conditions are uniform (Case II) or nonuniform (Case III). Due to the ribosomal queuing that occurs along the length of the mRNA at high polysome size (see 13 for more discussion), the effective elongation rate constants at positions spaced one-ribosome-length apart are approximately equal to the translocation rate constant, k₈ (see 13 for more discussion).

To investigate translation properties occurring in the regimes of polysome sizes associated with optimum rates, we obtained the elongation step control coefficients of each sequence at its respective optimum translation rate (Fig. 6). Similar to previous results 13, at the optimum rate the kinetics are completely elongation-limited, with . The control over rate is dominated by segments of codons that have high elongation step control coefficients (Fig. 8, results shown only for sequences producing highest and lowest optimum translation rates). For all of the polysome sizes that the translation kinetics are completely elongation-limited, we observe that the configuration of elongation step control coefficients does not change, and therefore the segments of rate-limiting codons do not change.

Display large version of this figure

Figure 8

Elongation step control coefficients, , with respect to sequence position under Case III conditions.

The positions of rate-limiting codon segments are expected because they correspond to segments of high translation time (Fig. 9). We define the translation time of the codon segments to be

(27)

Display large version of this figure

Figure 9

(A) Segment translation times, , with respect to sequence position for one of the sequences used in this study (similar results are observed for the other sequences). The thin line represents the segment translation times with U_n,r=1 for all codons in the sequence, while the thick line represents the segment translation times with U_n,r values corresponding to the ribosome distribution at the optimum translation rate. (B) Elongation step control coefficients, , with respect to sequence position under Case III conditions for the same sequence used in