April 2018, 15(2): 507-522. doi: 10.3934/mbe.2018023

Formulation of the protein synthesis rate with sequence information

1. 

Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, China

2. 

Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, China

* Corresponding author: Jinzhi Lei

Received  June 04, 2016 Accepted  March 16, 2017 Published  June 2017

Fund Project: This work is supported by National Natural Science Foundation of China (91430101 and 11272169)

Translation is a central biological process by which proteins are synthesized from genetic information contained within mRNAs. Here, we investigate the kinetics of translation at the molecular level by a stochastic simulation model. The model explicitly includes RNA sequences, ribosome dynamics, the tRNA pool and biochemical reactions involved in the translation elongation. The results show that the translation efficiency is mainly limited by the available ribosome number, translation initiation and the translation elongation time. The elongation time is a log-normal distribution, with the mean and variance determined by the codon saturation and the process of aa-tRNA selection at each codon binding site. Moreover, our simulations show that the translation accuracy exponentially decreases with the sequence length. These results suggest that aa-tRNA competition is crucial for both translation elongation, translation efficiency and the accuracy, which in turn determined the effective protein production rate of correct proteins. Our results improve the dynamical equation of protein production with a delay differential equation that is dependent on sequence information through both the effective production rate and the distribution of elongation time.

Citation: Wenjun Xia, Jinzhi Lei. Formulation of the protein synthesis rate with sequence information. Mathematical Biosciences & Engineering, 2018, 15 (2) : 507-522. doi: 10.3934/mbe.2018023
References:
[1]

M. M. BabuN. M. LuscombeL. AravindM. Gerstein and S. A. Teichmann, Structure and evolution of transcriptional regulatory networks, Curr. Opin. Struct. Biol., 14 (2004), 283-291. doi: 10.1016/j.sbi.2004.05.004.

[2]

G. CannarozziN. N. SchraudolphM. FatyP. von RohrM. T. FribergA. C. RothP. GonnetG. Gonnet and Y. Barral, A role for codon order in translation dynamics, Cell, 141 (2010), 355-367. doi: 10.1016/j.cell.2010.02.036.

[3]

D. ChuD. J. Barnes and T. von der Haar, The role of tRNA and ribosome competition in coupling the expression of different mRNAs in saccharomyces cerevisiae, Nucleic. Acids. Res., 39 (2011), 6705-6714. doi: 10.1093/nar/gkr300.

[4]

L. J. CoreA. L. MartinsC. G. DankoC. T. WatersA. Siepel and J. T. Lis, Analysis of nascent RNA identifies a unified architecture of initiation regions at mammalian promoters and enhancers, Nat. Genet., 46 (2014), 1311-1320. doi: 10.1038/ng.3142.

[5]

H. DongL. Nilsson and C. G. Kurland, Co-variation of tRNA abundance and codon usage in Escherichia coli at different growth rates, J. Mol. Biol., 260 (1996), 649-663. doi: 10.1006/jmbi.1996.0428.

[6]

A. FluittE. Pienaar and H. Viljoen, Ribosome kinetics and aa-tRNA competition determine rate and fidelity of peptide synthesis, Comput. Biol. Chem., 31 (2007), 335-346. doi: 10.1016/j.compbiolchem.2007.07.003.

[7]

D. T. Gilliespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem., 81 (1977), 2340-2361. doi: 10.1021/j100540a008.

[8]

K. B. Gromadski and M. V. Rodnina, Kinetic determinants of high-fidelity tRNA discrimination on the ribosome, Mol. Cell, 13 (2004), 191-200. doi: 10.1016/S1097-2765(04)00005-X.

[9]

M. GuttmanP. RussellN. T. IngoliaJ. S. Weissman and E. S. Lander, Ribosome profiling provides evidence that large noncoding RNAs do not encode proteins, Cell, 154 (2013), 240-251. doi: 10.1016/j.cell.2013.06.009.

[10]

N. T. IngoliaS. GhaemmaghamiJ. R. Newman and J. S. Weissman, Genome-wide analysis in vivo of translation with nucleotide resolution using ribosome profiling, Science, 324 (2009), 218-223. doi: 10.1126/science.1168978.

[11]

N. T. IngoliaL. F. Lareau and J. S. Weissman, Ribosome profiling of mouse embryonic stem cells reveals the complexity and dynamics of mammalian proteomes, Cell, 147 (2011), 789-802. doi: 10.1016/j.cell.2011.10.002.

[12]

R. J. JacksonC. U. Hellen and T. V. Pestova, The mechanism of eukaryotic translation initiation and principles of its regulation, Nat. Rev. Mol. Cell Biol., 11 (2010), 113-127. doi: 10.1038/nrm2838.

[13]

G.-W. Li and X. S. Xie, Central dogma at the single-molecule level in living cells, Nature, 475 (2011), 308-315. doi: 10.1038/nature10315.

[14]

E. LimpertW. Stahel and M. Abbt, Log-normal distributions across the sciences: Keys and clues, BioScience, 51 (2001), 341-352.

[15]

Y. MaoH. LiuY. Liu and S. Tao, Deciphering the rules by which dynamics of mRNA secondary structure affect translation efficiency in saccharomyces cerevisiae, Nucleic. Acids. Res., 42 (2014), 4813-4822. doi: 10.1093/nar/gku159.

[16]

N. MitaraiK. Sneppen and S. Pedersen, Ribosome collisions and translation efficiency: Optimization by codon usage and mRNA destabilization, J. Mol. Biol., 382 (2008), 236-245. doi: 10.1016/j.jmb.2008.06.068.

[17]

J. Ninio, Ribosomal kinetics and accuracy: sequence engineering to the rescue, J. Mol. Biol., 422 (2012), 325-327. doi: 10.1016/j.jmb.2012.07.002.

[18]

J. B. Plotkin and G. Kudla, Synonymous but not the same: The causes and consequences of codon bias, Nat. Rev. Genet., 12 (2010), 32-42. doi: 10.1038/nrg2899.

[19]

S. ProshkinA. R. RahmouniA. Mironov and E. Nudler, Cooperation between translating ribosomes and RNA polymerase in transcription elongation, Science, 328 (2010), 504-508. doi: 10.1126/science.1184939.

[20]

A. SavelsberghV. KatuninD. MohrF. PeskeM. Rodnina and W. Wintermeyer, An elongation factor G-induced ribosome rearrangement precedes tRNA-mRNA translocation, Mol. Cell, 11 (2003), 1517-1523. doi: 10.1016/S1097-2765(03)00230-2.

[21]

P. ShahY. DingM. NiemczykG. Kudla and J. B. Plotkin, Rate-limiting steps in yeast protein translation, Cell, 153 (2013), 1589-1601. doi: 10.1016/j.cell.2013.05.049.

[22]

P. Shah and M. A. Gilchrist, Explaining complex codon usage patterns with selection for translational efficiency, mutation bias, and genetic drift, Proc. Natl. Acad. Sci. USA, 108 (2011), 10231-10236. doi: 10.1073/pnas.1016719108.

[23]

M. Siwiak and P. Zielenkiewicz, A comprehensive, quantitative, and genome-wide model of translation, PLoS Comput. Biol., 6 (2010), e1000865. doi: 10.1016/0006-291X(79)91600-0.

[24]

S. S. Sommer and N. A. Rin, The lognormal distribution fits the decay profile of eukaryotic mRNA, Biochem Biophys Res Commun, 90 (1979), 135-141. doi: 10.1016/0006-291X(79)91600-0.

[25]

T. TianK. BurrageP. M. Burrage and M. Carletti, Stochastic delay differential equations for genetic regulatory networks, J. Comput. Appl. Math., 205 (2007), 696-707. doi: 10.1016/j.cam.2006.02.063.

[26]

T. TullerA. CarmiK. VestsigianS. NavonY. DorfanJ. ZaborskeT. PanO. DahanI. Furman and Y. Pilpel, An evolutionarily conserved mechanism for controlling the efficiency of protein translation, Cell, 141 (2010), 344-354. doi: 10.1016/j.cell.2010.03.031.

[27]

T. TullerY. Y. WaldmanM. Kupiec and E. Ruppin, Translation efficiency is determined by both codon bias and folding energy, Proc. Natl. Acad.Sci. USA, 107 (2010), 3645-3650. doi: 10.1073/pnas.0909910107.

[28]

G. von Heijne, Membrane-protein topology, Nat. Rev. Mol. Cell Biol., 7 (2006), 909-918. doi: 10.1038/nrm2063.

[29]

X. S. XieP. J. ChoiG.-W. LiN. K. Lee and G. Lia, Single-molecule approach to molecular biology in living bacterial cells, Annual review of biophysics, 37 (2008), 417-444. doi: 10.1146/annurev.biophys.37.092607.174640.

[30]

L. M. y Terán-RomeroM. Silber and V. Hatzimanikatis, The origins of time-delay in template biopolymerization processes, PLoS Comput. Biol., 6 (2010), e1000726, 15pp. doi: 10.1371/journal.pcbi.1000726.

[31]

E. Zavala and T. T. Marquez-Lago, Delays induce novel stochastic effects in negative feedback gene circuits, Biophys. J., 106 (2014), 467-478. doi: 10.1016/j.bpj.2013.12.010.

show all references

References:
[1]

M. M. BabuN. M. LuscombeL. AravindM. Gerstein and S. A. Teichmann, Structure and evolution of transcriptional regulatory networks, Curr. Opin. Struct. Biol., 14 (2004), 283-291. doi: 10.1016/j.sbi.2004.05.004.

[2]

G. CannarozziN. N. SchraudolphM. FatyP. von RohrM. T. FribergA. C. RothP. GonnetG. Gonnet and Y. Barral, A role for codon order in translation dynamics, Cell, 141 (2010), 355-367. doi: 10.1016/j.cell.2010.02.036.

[3]

D. ChuD. J. Barnes and T. von der Haar, The role of tRNA and ribosome competition in coupling the expression of different mRNAs in saccharomyces cerevisiae, Nucleic. Acids. Res., 39 (2011), 6705-6714. doi: 10.1093/nar/gkr300.

[4]

L. J. CoreA. L. MartinsC. G. DankoC. T. WatersA. Siepel and J. T. Lis, Analysis of nascent RNA identifies a unified architecture of initiation regions at mammalian promoters and enhancers, Nat. Genet., 46 (2014), 1311-1320. doi: 10.1038/ng.3142.

[5]

H. DongL. Nilsson and C. G. Kurland, Co-variation of tRNA abundance and codon usage in Escherichia coli at different growth rates, J. Mol. Biol., 260 (1996), 649-663. doi: 10.1006/jmbi.1996.0428.

[6]

A. FluittE. Pienaar and H. Viljoen, Ribosome kinetics and aa-tRNA competition determine rate and fidelity of peptide synthesis, Comput. Biol. Chem., 31 (2007), 335-346. doi: 10.1016/j.compbiolchem.2007.07.003.

[7]

D. T. Gilliespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem., 81 (1977), 2340-2361. doi: 10.1021/j100540a008.

[8]

K. B. Gromadski and M. V. Rodnina, Kinetic determinants of high-fidelity tRNA discrimination on the ribosome, Mol. Cell, 13 (2004), 191-200. doi: 10.1016/S1097-2765(04)00005-X.

[9]

M. GuttmanP. RussellN. T. IngoliaJ. S. Weissman and E. S. Lander, Ribosome profiling provides evidence that large noncoding RNAs do not encode proteins, Cell, 154 (2013), 240-251. doi: 10.1016/j.cell.2013.06.009.

[10]

N. T. IngoliaS. GhaemmaghamiJ. R. Newman and J. S. Weissman, Genome-wide analysis in vivo of translation with nucleotide resolution using ribosome profiling, Science, 324 (2009), 218-223. doi: 10.1126/science.1168978.

[11]

N. T. IngoliaL. F. Lareau and J. S. Weissman, Ribosome profiling of mouse embryonic stem cells reveals the complexity and dynamics of mammalian proteomes, Cell, 147 (2011), 789-802. doi: 10.1016/j.cell.2011.10.002.

[12]

R. J. JacksonC. U. Hellen and T. V. Pestova, The mechanism of eukaryotic translation initiation and principles of its regulation, Nat. Rev. Mol. Cell Biol., 11 (2010), 113-127. doi: 10.1038/nrm2838.

[13]

G.-W. Li and X. S. Xie, Central dogma at the single-molecule level in living cells, Nature, 475 (2011), 308-315. doi: 10.1038/nature10315.

[14]

E. LimpertW. Stahel and M. Abbt, Log-normal distributions across the sciences: Keys and clues, BioScience, 51 (2001), 341-352.

[15]

Y. MaoH. LiuY. Liu and S. Tao, Deciphering the rules by which dynamics of mRNA secondary structure affect translation efficiency in saccharomyces cerevisiae, Nucleic. Acids. Res., 42 (2014), 4813-4822. doi: 10.1093/nar/gku159.

[16]

N. MitaraiK. Sneppen and S. Pedersen, Ribosome collisions and translation efficiency: Optimization by codon usage and mRNA destabilization, J. Mol. Biol., 382 (2008), 236-245. doi: 10.1016/j.jmb.2008.06.068.

[17]

J. Ninio, Ribosomal kinetics and accuracy: sequence engineering to the rescue, J. Mol. Biol., 422 (2012), 325-327. doi: 10.1016/j.jmb.2012.07.002.

[18]

J. B. Plotkin and G. Kudla, Synonymous but not the same: The causes and consequences of codon bias, Nat. Rev. Genet., 12 (2010), 32-42. doi: 10.1038/nrg2899.

[19]

S. ProshkinA. R. RahmouniA. Mironov and E. Nudler, Cooperation between translating ribosomes and RNA polymerase in transcription elongation, Science, 328 (2010), 504-508. doi: 10.1126/science.1184939.

[20]

A. SavelsberghV. KatuninD. MohrF. PeskeM. Rodnina and W. Wintermeyer, An elongation factor G-induced ribosome rearrangement precedes tRNA-mRNA translocation, Mol. Cell, 11 (2003), 1517-1523. doi: 10.1016/S1097-2765(03)00230-2.

[21]

P. ShahY. DingM. NiemczykG. Kudla and J. B. Plotkin, Rate-limiting steps in yeast protein translation, Cell, 153 (2013), 1589-1601. doi: 10.1016/j.cell.2013.05.049.

[22]

P. Shah and M. A. Gilchrist, Explaining complex codon usage patterns with selection for translational efficiency, mutation bias, and genetic drift, Proc. Natl. Acad. Sci. USA, 108 (2011), 10231-10236. doi: 10.1073/pnas.1016719108.

[23]

M. Siwiak and P. Zielenkiewicz, A comprehensive, quantitative, and genome-wide model of translation, PLoS Comput. Biol., 6 (2010), e1000865. doi: 10.1016/0006-291X(79)91600-0.

[24]

S. S. Sommer and N. A. Rin, The lognormal distribution fits the decay profile of eukaryotic mRNA, Biochem Biophys Res Commun, 90 (1979), 135-141. doi: 10.1016/0006-291X(79)91600-0.

[25]

T. TianK. BurrageP. M. Burrage and M. Carletti, Stochastic delay differential equations for genetic regulatory networks, J. Comput. Appl. Math., 205 (2007), 696-707. doi: 10.1016/j.cam.2006.02.063.

[26]

T. TullerA. CarmiK. VestsigianS. NavonY. DorfanJ. ZaborskeT. PanO. DahanI. Furman and Y. Pilpel, An evolutionarily conserved mechanism for controlling the efficiency of protein translation, Cell, 141 (2010), 344-354. doi: 10.1016/j.cell.2010.03.031.

[27]

T. TullerY. Y. WaldmanM. Kupiec and E. Ruppin, Translation efficiency is determined by both codon bias and folding energy, Proc. Natl. Acad.Sci. USA, 107 (2010), 3645-3650. doi: 10.1073/pnas.0909910107.

[28]

G. von Heijne, Membrane-protein topology, Nat. Rev. Mol. Cell Biol., 7 (2006), 909-918. doi: 10.1038/nrm2063.

[29]

X. S. XieP. J. ChoiG.-W. LiN. K. Lee and G. Lia, Single-molecule approach to molecular biology in living bacterial cells, Annual review of biophysics, 37 (2008), 417-444. doi: 10.1146/annurev.biophys.37.092607.174640.

[30]

L. M. y Terán-RomeroM. Silber and V. Hatzimanikatis, The origins of time-delay in template biopolymerization processes, PLoS Comput. Biol., 6 (2010), e1000726, 15pp. doi: 10.1371/journal.pcbi.1000726.

[31]

E. Zavala and T. T. Marquez-Lago, Delays induce novel stochastic effects in negative feedback gene circuits, Biophys. J., 106 (2014), 467-478. doi: 10.1016/j.bpj.2013.12.010.

Figure 1.  Kinetic scheme of RNA translation. Re-drawn from [6]
Figure 2.  Translation kinetics of a single mRNA sequence. (a) Positions of each ribosome on the sequence. (b) Numbers of protein products. The black solid line represents all protein products, and the red dashed line represents the correctly translated proteins (no incorrect amino acid added by near-cognate aa-tRNAs). Here, the sample sequence is the gene YAL003W from the SGD yeast coding sequence, with a sequence length $L = 621 nt$. The simulation time, on Mac Pro with $2\times 3.06$ GHz 6-Core Intel Xeon and 16 GB memory, was about 3 min. Parameters are $R=20$ and $F=0.03$. For other parameters refer to Table 1
Figure 3.  Distribution of the elongation time per codon during the translation of YAL003W. All parameters are the same as described in Fig. 2. The red curve is the fit with the normal distribution $\ln \mathcal{N}(-1.6, 1.69)$
Figure 4.  Dependence of the $ETC$ of yeast coding sequences on tRNA usage. Dots represent the mean (upper panel) and variance (bottom panel) of the logarithm of $ETC$ with cognate tRNA usage $F_{\mathrm{cog}}$, near-cognate tRNA usage $F_{\mathrm{near}}$ and non-cognate tRNA usage $F_{\mathrm{non}}$. Dashed lines show the linear fitting. Simulations of 1000 yeast coding sequences are shown; each dot corresponds to one sequence. All parameters are the same as described in Fig. 2
Figure 5.  Dependence of the elongation time on the available ribosome number $R$. (a) Average $ETC$ versus $R$. (b) Ribosome distance (in codons) versus $R$. The sequence and parameters are the same as described in Fig. 2
Figure 6.  Dependence of the $ETC$ on total tRNA number represented by the factor $F$. The mean (left hand ordinate, blue circles connected with a dashed line) and variance (right hand ordinate, red triangles connected with a dotted line) of the logarithm of $ETC$ are shown as a function of the factor $F$. The sequence and parameters are the same as described in Fig. 2
Figure 7.  Dependence of translation efficiency on the maximum number of available ribosomes $R$. The dashed lines represent show the two-phase dependence following Eq. 11. The sequence and parameters are the same as described in Fig. 2
Figure 8.  Translation kinetics. (a) Translation efficiency versus sequence length for 1000 yeast coding sequences. Red line shows the fitting with $TE = \dfrac{0.195}{1+0.0033 n}$. (b) Translation accuracy versus sequence length for 1000 yeast coding genes. Red line shows the fitting with $e^{-0.0042 n}$. Here, $n=L/3$ represents the protein chain length. Data were obtained from the simulation shown in Fig. 4
Figure 9.  Sensitivity analysis of translation efficiency. Bars show changes in the logarithm of translation efficiencies induced by changes in a single parameter $\ln(TE^*/TE_0)$, where $TE^*$ and $TE_0$ represent the $TE$ for modified and default parameters, respectively. Blue bars correspond to the increase of a parameter by $10\%$, and yellow bars correspond to the decrease of a parameter by $10\%$. For parameters refer to Table 1, the parameters $\mathrm{ke02}$, $\mathrm{ke3}$, $\mathrm{ke5}$, $\mathrm{ke7}$, and $\mathrm{keT}$ for values of $\mathrm{k02}$, $\mathrm{k3}$, $\mathrm{k5}$, $\mathrm{k7}$, and $\mathrm{kT}$ of near-cognate tRNAs (second column in Table 1), respectively, and $\mathrm{kn01}$ for the parameter $\mathrm{k01}$ of the non-cognate tRNAs (third column in Table 1). The sequence and default parameters are the same as described in Fig. 2
Figure 10.  $ETC$ of the translation for different samples. Distributions of the mean and variance of the logarithm of $ETC$ for yeast coding RNAs (a), yeast noncoding RNAs (b), human coding RNAs (c) and human noncoding RNAs (d). Here, the results of 500 random sequences with lengths of $200 nt < L< 1000 nt$ for each sample are shown. Red stars show the average values for each sample; the values are provided in the table. The parameters are $R=20, F=0.03$; for other parameters, refer to Table 1
Table 1.  Values of kinetic rate constants ($s^{-1}$) (refer to [6])
Parameters Values Cognate Near-cognate Non-cognate
K0.03---
k1-1401402000
k01-8585-
k2-190190-
k02-0.2380-
k3-2600.4-
kG-10001000-
k4-10001000-
k5-100060-
k7-601000-
kp-200200-
kT-2020-
Parameters Values Cognate Near-cognate Non-cognate
K0.03---
k1-1401402000
k01-8585-
k2-190190-
k02-0.2380-
k3-2600.4-
kG-10001000-
k4-10001000-
k5-100060-
k7-601000-
kp-200200-
kT-2020-
Table 2.  tRNA pool composition (refer to [5,6]). Also refer to [6] for the anti-codons for the tRNAs
tRNA Molecules/cell tRNA Molecules/cell tRNA Molecules/cell
Ala13250His639Pro3581
Ala2617Ile11737Sec219
Arg24752Ile21737Ser11296
Arg3639Leu14470Ser2344
Arg4867Leu2943Ser31408
Arg5420Leu3666Ser5764
Asn1193Leu41913Thr1104
Asp12396Leu51031Thr2541
Cys1587Lys1924Thr31095
Gln1764Met f11211Thr4916
Gln2881Met f2715Trp943
Glu24717Met m706Tyr1769
Gly11068Phe1037Tyr21261
Gly21068Pro1900Val13840
Gly34359Pro2720Val2A630
Val2B635
tRNA Molecules/cell tRNA Molecules/cell tRNA Molecules/cell
Ala13250His639Pro3581
Ala2617Ile11737Sec219
Arg24752Ile21737Ser11296
Arg3639Leu14470Ser2344
Arg4867Leu2943Ser31408
Arg5420Leu3666Ser5764
Asn1193Leu41913Thr1104
Asp12396Leu51031Thr2541
Cys1587Lys1924Thr31095
Gln1764Met f11211Thr4916
Gln2881Met f2715Trp943
Glu24717Met m706Tyr1769
Gly11068Phe1037Tyr21261
Gly21068Pro1900Val13840
Gly34359Pro2720Val2A630
Val2B635
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