Stochastic kinetics of ribosomes: single motor properties and collective behavior

arXiv:0903.2608v3 [physics.bio-ph] 12 Jul 2009 Stochastic kinetics of ribosomes: single motor properties and collective behavior Ashok Garai,1 Debanjan Chowdhury,1 Debashish Chowdhury∗,1 and T.V. Ramakrishnan2, 3 1Department of Physics, Indian Institute of Technology, Kanpur 208016, India. 2Department of Physics, Banaras Hindu University, Varanasi 221005, India. 3Department of Physics, Indian Institute of Science, Bangalore 560012, India. Synthesis of protein molecules in a cell are carried out by ribosomes. A ribosome can be regarded as a molecular motor which utilizes the input chemical energy to move on a messenger RNA (mRNA) track that also serves as a template for the polymerization of the corresponding protein. The forward movement, however, is characterized by an alternating sequence of translocation and pause. Using a quantitative model, which captures the mechanochemical cycle of an individual ribosome, we derive an exact analytical expression for the distribution of its dwell times at the successive positions on the mRNA track. Inverse of the average dwell time satisfies a “Michaelis-Menten-like” equation and is consistent with the general formula for the average velocity of a molecular motor with an unbranched mechano-chemical cycle. Extending this formula appropriately, we also derive the exact force-velocity relation for a ribosome. Often many ribosomes simultaneously move on the same mRNA track, while each synthesizes a copy of the same protein. We extend the model of a single ribosome by incorporating steric exclusion of different individuals on the same track. We draw the phase diagram of this model of ribosome traffic in 3-dimensional spaces spanned by experimentally controllable parameters. We suggest new experimental tests of our theoretical predictions. PACS numbers: 87.16.Ac 89.20.-a I. INTRODUCTION Ribosome is one of the largest and most complex in- tracellular cyclic molecular machines [1, 2, 3, 4] and it plays a crucial role in gene expression [5]. It synthesizes a protein molecule, which is a hetero polymer of amino acid subunits, using a messenger RNA (mRNA) as the corresponding template; this process is called translation (of the genetic message). Monomeric subunits of RNA are nucleotides and triplets of nucleotides constitute a codon. The dictionary of translation relates each type of possible codon with one species of amino acid. Thus, the sequence of amino acids on a protein is dictated by the sequence of codons on the corresponding template mRNA. The polymerization of protein takes places in three stages which are identified as initiation, elongation (of the protein) and termination. In this paper we focus almost exclusively on the elongation stage. A ribosome is often treated as a molecular motor for which the mRNA template also serves as a track. In each step it moves forward on its track by one codon by consuming chemical fuel [e.g., two guanosine tri- phosphate(GTP) molecules]. Simultaneously, in each step, it also elongates the protein by adding an amino acid; the correct sequence of the amino acids required for polymerizing a protein is dictated by the codon se- quence on the mRNA template. Therefore, it may be more appropriate to regard a ribosome as a mobile work- shop that provides a platform for operation of several tools in a well coordinated manner. Our main aim is to predict the effects of the mechano-chemical cycle of ∗Corresponding author(E-mail: debch@iitk.ac.in) individual ribosomes, in the elongation stage, on their experimentally measurable physical properties. We first focus on the single-ribosome properties which character- ize their stochastic movement on the track in the absence of inter-ribosome interactions. Then we consider the ad- ditional effects of the steric interactions of the ribosomes and those of the rates of initiation and termination of translation on the collective spatio-temporal organization of the ribosomes on a track. The stochastic forward movement of a ribosome is characterized by an alternating sequence of pause and translocation. The sum of the durations of a pause and the following translocation defines the time of a dwell at the corresponding codon. Recently, using an inge- nious method, the distribution f(t) of the dwell times of a ribosome has been measured [6]. We present a system- atic derivation of this distribution from a detailed kinetic theory of translation which incorporates the mechano- chemical cycle of individual ribosomes. The exact analytical expression for f(t) which we de- rive here is, in general, a superposition of several expo- nentials. On the other hand, it has been claimed [6] that difference of two exponentials fit the experimentally mea- sured f(t) very well. We reconcile these two observations by identifying the parameter regime where our theoret- ically derived f(t) is, indeed, well approximated by dif- ference of two exponentials [7, 8, 9, 10, 11]. Moreover, we show that ⟨t⟩−1, inverse of the mean dwell time, sa

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