Comment on 'Length-dependent translation of messenger RNA by ribosomes'

arXiv:1109.3594v2 [physics.bio-ph] 10 Oct 2011 Comment on “Length-dependent translation of messenger RNA by ribosomes” Yunxin Zhang Shanghai Key Laboratory for Contemporary Applied Mathematics, Laboratory of Mathematics for Nonlinear Science, Centre for Computational Systems Biology, and School of Mathematical Sciences, Fudan University, Shanghai 200433, China. Email: xyz@fudan.edu.cn. In the recent paper of Valleriani et al [Phys. Rev. E 83, 042903 (2011)], a simple model for describing the translation of messenger RNA (mRNA) by ribosomes is presented, and an expression of the translational ratio r, defined as the ratio of translation rate ωtl of protein from mRNA to degradation rate ωp of protein, is obtained. The key point to get this ratio r is to get the translation rate ωtl. In the study of Valleriani et al, ωtl is assumed to be the mean value of measured translation rate, i.e. the mean value of ratio of the translation number of protein to the lifetime of mRNA. However, in experiments different methods might be used to get ωtl. Therefore, for the sake of future application of their model to more experimental data analysis, in this comment three methods to get the translation rate ωtl, and consequently the translational ratio r, are provided. Based on one of the methods which might be employed in most of the experiments, we find that the translational ratio r decays exponentially with the length of mRNA in prokaryotic cells, and decays reciprocally with the length of mRNA in eukaryotic cells. This result is slight different from that obtained in Valleriani’s study. In recent paper [1], Valleriani et al presented a sim- ple model to describe the length-dependent translation properties of messager RNA (mRNA). In their model, the mRNA degradation process is assumed to be gov- erned by rate ωr, i.e. the probability density of lifetime t of an intact mRNA is φU = ωr exp(−ωrt). (1) Meanwhile, the rate of ribosome entering the coding re- gion of mRNA is assumed to be ωon, and the degradation rate of protein is denoted by ωp. To discuss the mRNA length-dependent properties of the translation to protein by ribosomes, in [1] the expres- sion of translational ratio, defined as r = ωtl/ωp, (2) is obtained for translations in both prokaryotic and eu- karyotic cells. Where ωtl is the translation rate from mRNA to protein by ribosomes. Since the protein degra- dation rate ωp is independent of mRNA, the essential point to analyze the mRNA length-dependent proper- ties of the translational ratio r, is to get the expression of translation rate ωtl. Experimentally, there might be three methods to get ωtl: (I): ωtl is obtained as the mean value of the measured translation rate f(t) from mRNA to protein, i.e., ωtl = ⟨f(t)⟩= Z ∞ 0 f(t)φU(t)dt, (3) where f(t) = N(t)/T (t), N(t) is the mean number of proteins that the mRNA will synthesize if degradation occurs at time t, and T (t) is the lifetime of a mRNA before completely degraded (t is the lifetime of an intact mRNA). (II): ωtl is obtained as the ratio of mean number ⟨N(t)⟩ of proteins translated from one mRNA to the mean life- time ⟨T (t)⟩of a mRNA, i.e., ωtl = ⟨N(t)⟩/ ⟨T (t)⟩. (4) (III): ωtl is obtained as the reciprocal of the mean dura- tion time of translating one protein from mRNA, ωtl = 1 ⟨1/f(t)⟩= 1 ⟨T (t)/N(t)⟩. (5) One can easily show that, for the mRNA transla- tion problem discussed in [1], T (t) = t, N(t) = θ(t − 2 tpro L )ωon(t −tpro L ) for translation in prokaryotic cells, and T (t) = t + teu L , N(t) = ωont for translation in eukary- otic cells. Where θ(t) is Heaviside function, i.e. θ(t) = 1 for t > 0 and θ(t) = 0 for t < 0, tpro L = L/vpro and teu L = L/veu are the time taken by ribosomes to reach the end of mRNA, L is the length of mRNA, vpro and veu are the average velocities of ribosome along mRNA in prokaryotic and eukaryotic cells respectively. Intuitively, method (I) is reasonable, and actually this method is used in [1]. On the other hand, method (III) is usually employed for some mathematical problems. For example, to get the mean translation rate ω of a process which includes two sub-processes with rate ω1 and ω2, one usually does the following calculations ω = 1 ⟨T ⟩= 1 ⟨T1⟩+ ⟨T2⟩= 1 1/ω1 + 1/ω2 = ω1ω2 ω1 + ω2 . (6) It should be pointed out that, if the method (III) is employed to get ωtl, the experimental samples with no protein synthesized, i.e. samples for N(t) = 0, should be discarded to avoid the infinite waiting time cases. Cor- respondingly, in the theoretical calculation in Eq. (5), the average ⟨T (t)/N(t)⟩should be done for only large enough time t ≥tlim which satisfies N(t) ≥1, and the probability density φU should be changed accord- ingly, ˆφU(t) = φU(t)/ R ∞ tlim φU(t)dt = ωr exp[−ωr(t−tlim)] for t ≥tlim. In our numerical calculations, we use tpro lim = tpro L + 1/ωon for translation in prokaryotic cells, and teu lim = 1/ωon for translation in eukaryotic cells. Meanwhile, the simple method (II) is often used in the experimental data sta

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