# American Institute of Mathematical Sciences

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
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##### References:
Kinetic scheme of RNA translation. Re-drawn from [6]
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
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 $\ln \mathcal{N}(-1.6, 1.69)$
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
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
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
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
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
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
$ETC$ 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
Values of kinetic rate constants ($s^{-1}$) (refer to [6])
 Parameters Values Cognate Near-cognate Non-cognate K 0.03 - - - k1 - 140 140 2000 k01 - 85 85 - k2 - 190 190 - k02 - 0.23 80 - k3 - 260 0.4 - kG - 1000 1000 - k4 - 1000 1000 - k5 - 1000 60 - k7 - 60 1000 - kp - 200 200 - kT - 20 20 -
 Parameters Values Cognate Near-cognate Non-cognate K 0.03 - - - k1 - 140 140 2000 k01 - 85 85 - k2 - 190 190 - k02 - 0.23 80 - k3 - 260 0.4 - kG - 1000 1000 - k4 - 1000 1000 - k5 - 1000 60 - k7 - 60 1000 - kp - 200 200 - kT - 20 20 -
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 Ala1 3250 His 639 Pro3 581 Ala2 617 Ile1 1737 Sec 219 Arg2 4752 Ile2 1737 Ser1 1296 Arg3 639 Leu1 4470 Ser2 344 Arg4 867 Leu2 943 Ser3 1408 Arg5 420 Leu3 666 Ser5 764 Asn 1193 Leu4 1913 Thr1 104 Asp1 2396 Leu5 1031 Thr2 541 Cys 1587 Lys 1924 Thr3 1095 Gln1 764 Met f1 1211 Thr4 916 Gln2 881 Met f2 715 Trp 943 Glu2 4717 Met m 706 Tyr1 769 Gly1 1068 Phe 1037 Tyr2 1261 Gly2 1068 Pro1 900 Val1 3840 Gly3 4359 Pro2 720 Val2A 630 Val2B 635
 tRNA Molecules/cell tRNA Molecules/cell tRNA Molecules/cell Ala1 3250 His 639 Pro3 581 Ala2 617 Ile1 1737 Sec 219 Arg2 4752 Ile2 1737 Ser1 1296 Arg3 639 Leu1 4470 Ser2 344 Arg4 867 Leu2 943 Ser3 1408 Arg5 420 Leu3 666 Ser5 764 Asn 1193 Leu4 1913 Thr1 104 Asp1 2396 Leu5 1031 Thr2 541 Cys 1587 Lys 1924 Thr3 1095 Gln1 764 Met f1 1211 Thr4 916 Gln2 881 Met f2 715 Trp 943 Glu2 4717 Met m 706 Tyr1 769 Gly1 1068 Phe 1037 Tyr2 1261 Gly2 1068 Pro1 900 Val1 3840 Gly3 4359 Pro2 720 Val2A 630 Val2B 635
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