Spatial Fluctuations Affect the Dynamics of Motor Proteins Rahul Kumar Das and Anatoly B. Kolomeisky* Department of Chemistry, Rice UniVersity, Houston, Texas 77005-1892 ReceiVed: February 1, 2008; ReVised Manuscript ReceiVed: April 8, 2008 Motor proteins are active biological molecules that perform their functions by converting chemical energy into mechanical work. They move unidirectionally along rigid protein ﬁlaments or DNA and RNA molecules in discrete steps by hydrolyzing ATP (adenosine triphsophate) or related energy-rich compounds. Recent single-molecule experiments have shown that motor proteins experience signiﬁcant spatial ﬂuctuations during its motion, leading to broad step-size distributions. The effect of these spatial ﬂuctuations is analyzed explicitly by considering discrete-state stochastic models that allow us to compute exactly all dynamic properties. It is shown that for symmetric spatial ﬂuctuations there is no change in mean velocities for weak external forces, while dispersions and stall forces are strongly affected at all conditions. These results are illustrated by several simple examples. Our method is also applied to analyze the effect of step-size ﬂuctuations on dynamics of myosin V motor proteins. It is argued that spatial ﬂuctuations might be used to control and regulate the dynamics of motor proteins. 1. Introduction Several classes of active enzymatic molecules that produce mechanical work by utilizing energy of different biochemical processes are known as motor proteins, or molecular motors. 1–4 These molecules, such as kinesins, dyneins, myosins, DNA and RNA polymerases, helicases and many others, play important roles in a variety of biological processes that include cellular transport, cell division, muscle contraction, and genetic transcription. 1–3 They typically translocate in a linear fashion along rigid protein ﬁlaments or DNA and RNA molecules, and their motion is fueled by the energy of hydrolysis of ATP (adenosine triphosphate) or related compounds. However, mechanisms of coupling between biochemical transitions and mechanical transformations in motor proteins are still not well understood. 3,4 A large progress in understanding mechanisms of motor protein dynamics has been achieved in the past decade with the development of single-molecule experimental methods. 3–23 These investigations have revealed dynamic properties of molecular motors, such as velocities, dispersions, run lengths, dwell times, and stall forces, at different conditions for individual single protein molecules. It was shown that motor proteins can exert signiﬁcant forces during their motion, and there are large ﬂuctuations and variability in the dynamic properties. In addition, functioning of motor proteins includes multiple states and conformations that are related via complex biochemical pathways. Signiﬁcant advances in experimental investigations of motor proteins, which enabled the description of dynamics and biochemical transitions at the single-molecule level, have greatly stimulated theoretical discussions on the functioning of molec- ular motors. 3,4,24–31 Theoretical studies of motor proteins mostly involve two main directions: continuum ratchet models 24,29–31 and stochastic discrete-state models. 4,25–28 Current theoretical approaches can account for most available experimental obser- vations, and they provide a reasonable framework for under- standing mechanisms of molecular motor’s transport. 4 One of the most fascinating properties of motor proteins is a large variability and ﬂuctuations in dynamic properties. The precision of existing single-molecule experimental techniques allows to quantify these ﬂuctuations, indicating that they contain an important information about biochemical and biophysical processes in motor proteins. 8,10–23 Thus, the use of ﬂuctuations and variability data might provide a valuable tool for under- standing mechanisms of molecular motor’s functioning. How- ever, theoretical descriptions of these phenomena in motor proteins are rather very limited. 4,32 The ﬁrst simpliﬁed approach to take into account spatial ﬂuctuations has been presented in ref 27. Here, the upper bounds of the effect of ﬂuctuations on dynamics have been obtained by assuming (obviously, unreal- istically) that the myosin V molecules move via alternating long and short steps. Although the method was quite naive, it showed that ﬂuctuations might modify the dynamics only near the stalling force conditions where the precision of experimental measurements is not high. Recently, Shaevitz, Block, and Schnitzer 32 presented a ﬁrst analytical study of spatial ﬂuctuations in motor proteins step sizes. Using a moment-generating functions method, they calculated distribution functions for completion times that allowed them to analyze the effect of variability in the step size of motor proteins on their dynamics. Speciﬁcally, they consid- ered a randomness parameter r deﬁned as 4,6,32 r ) 2D dV (1) where D and V are mean dispersion and velocity of the motor protein molecule and d is the average step-size. This function provides a convenient measure of overall ﬂuctuations in molecular motors. It was shown that the randomness r can be written as a sum of two terms corresponding to ﬂuctuations in the step-size and due to the stochastic nature of enzymatic molecules. 32 Although this theoretical work provides a valuable description of ﬂuctuations, its application is restricted because of the assumption of irreversibility in biochemical transitions of motor proteins. Generally, all chemical reactions are revers- ible, and neglecting this property might lead to erroneous * tolya@rice.edu J. Phys. Chem. B 2008, 112, 11112–11121 11112 10.1021/jp800982b CCC: $40.75  2008 American Chemical Society Published on Web 08/08/2008