RNA polymerase (RNAP) is a mobile molecular workshop that polymerizes a RNA molecule by adding monomeric subunits one by one, while moving step by step on the DNA template itself. Here we develop a theoretical model by incorporating the steric interactions of the RNAPs and their mechanochemical cycles which explicitly captures the cyclical shape changes of each motor. Using this model, we explain not only the dependence of the average velocity of a RNAP on the externally applied load force, but also predict a {\it nonmotonic} variation of the average velocity on external torque. We also show the effect of steric interactions of the motors on the total rate of RNA synthesis. In principle, our predictions can be tested by carrying out {\it in-vitro} experiments which we suggest here.
Deep Dive into RNA polymerase motor on DNA track: effects of interactions, external force and torque.
RNA polymerase (RNAP) is a mobile molecular workshop that polymerizes a RNA molecule by adding monomeric subunits one by one, while moving step by step on the DNA template itself. Here we develop a theoretical model by incorporating the steric interactions of the RNAPs and their mechanochemical cycles which explicitly captures the cyclical shape changes of each motor. Using this model, we explain not only the dependence of the average velocity of a RNAP on the externally applied load force, but also predict a {\it nonmotonic} variation of the average velocity on external torque. We also show the effect of steric interactions of the motors on the total rate of RNA synthesis. In principle, our predictions can be tested by carrying out {\it in-vitro} experiments which we suggest here.
Molecular motors [1] in living cells are either proteins or macromolecular complexes made of proteins and ribonucleic acids (RNAs). Like their macroscopic counterparts, these motors perform mechanical work, while translocating on a filamentous track, by converting input energy which is often supplied as chemical energy [1,2,3]. In this paper we study a special class of motors, called RNA polymerase (RNAP) which play crucial roles in gene expression [4].
Transcription of a gene encoded in the sequence of nucleotides in a specific segment of a DNA is carried out by RNAP motors which treat the DNA as a template [5,6]. An RNAP is more like a mobile workshop that performs three functions simultaneously: (i) it decodes the genetic message encoded in the template DNA and selects the appropriate nucleotide, the monomeric subunit of RNA, as dictated by the template, (ii) it catalyzes the addition of the monomeric subunit thus selected to the growing RNA molecule, (iii) it steps forward by one nucleotide on its template without completely destabilizing the ternary complex consisting of the polymerase, the template DNA and the product RNA. The free energy released by the polymerization of the RNA molecule serves as the input energy for the driving the mechanical movements of the RNAP. Therefore, these enzymes are also regarded as molecular motors [7].
During the transcription of a gene, the collective movement of the RNAPs on the same DNA track is often referred to as RNAP traffic because of its superficial similarity with vehicular traffic [8,9]. The beginning and the end of the specific sequence corresponding to a gene are the analogs of the on-ramp and off-ramp of vehicular traffic on highways. The average number of RNAPs, which complete the synthesis of a RNA molecule per unit time interval can be identified as the flux in RNAP traffic. Note that flux is the product of the number density and average velocity of the motors. Thus, flux in RNAP traffic is identical to the average rate of synthesis of the corresponding RNA. Using the terminology of traffic science [9], we’ll call the relation between the flux and the number density of the motors as the fundamental diagram. The fundamental diagram is an important quantitative characteristic of traffic flow.
The dependence of the velocity of the motor on an externally imposed load (opposing) force is called the force-velocity relation which is one of the most important characteristics of a molecular motor. The force-velocity relation for RNAP motors have been measured by carrying out single molecule experiments [10,11]. However, to our knowledge, the response of an RNAP motor to an externally applied torque has not been investigated so far. The effects of steric interactions of the RNAP motors on their dynamics has been studied only in a few experiments [12,13,14,15,16]; but, none of these addressed the question of the nature of the overall spatio-temporal organization of the RNAP motors in RNAP traffic.
The traffic-like collective dynamics of cytoskeletal molecular motors [17,18,19,20,21,22] and that of ribosomes on mRNA tracks [23,24,25,26,27,28,29,30,31,32,33,34] have been investigated theoretically in the physics literature. However, so far, RNAP traffic has received far less attention [35,36,37] In this paper we develop a model that captures not only the steric interactions between the RNAPs, but also separately the biochemical reactions catalyzed by an RNAP and the cyclic shape changes it undergoes during each mechano-chemical cycle. This model may be regarded as a “unified” description in the sense that the same model describes the single RNAP properties (e.g., the force-velocity and torque-velocity relations) as well as the collective spatio-temporal organization (and rate of RNA synthesis).
The main stages in the polymerization of polynucleotides by the polymerase machines are common: (a) initiation: Once the polymerase encounters a specific sequence on the template that acts as a chemically coded start signal, it initiates the synthesis of the product. The RNAP, together with the DNA bubble and the growing RNA transcript, forms a “transcription elongation complex” (TEC). This stage is completed when the nascent product RNA becomes long enough to stabilize the TEC against dissociation from the template. (b) elongation: During this stage, the nascent product gets elongated by the addition of nucleotides; during elongation [38], each successful addition of a nucleotide to the elongating mRNA leads to a forward stepping of the RNAP. (c) termination: Normally, the process of synthesis is terminated, and the newly polymerized full length product molecule is released, when the polymerase encounters the terminator (or, stop) sequence on the template. In this paper we are interested mainly in the elongation of the mRNA transcripts.
FIG. 1: (Color online) A schematic depiction of the essential architectural feature of a RNAP that we capture in the model proposed i
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