Extracting the intrinsic kinetic information of biological molecule from its single-molecule kinetic data is of considerable biophysical interest. In this work, we theoretically investigate the feasibility of inferring single RNA's intrinsic kinetic parameters from the time series obtained by forced folding/unfolding experiment done in the light tweezer, where the molecule is flanked by long double-stranded DNA/RNA handles and tethered between two big beads. We first construct a coarse-grain physical model of the experimental system. The model has captured the major physical factors: the Brownian motion of the bead, the molecular structural transition, and the elasticity of the handles and RNA. Then based on an analytic solution of the model, a Bayesian method using Monte Carlo Markov Chain is proposed to infer the intrinsic kinetic parameters of the RNA from the noisy time series of the distance or force. Because the force fluctuation induced by the Brownian motion of the bead and the structural transition can significantly modulate the transition rates of the RNA, we prove that, this statistic method is more accurate and efficient than the conventional histogram fitting method in inferring the molecule's intrinsic parameters.
The current Single-molecule manipulation provides a novel approach to study the kinetics of single RNA. Different from many conventional experimental techniques, such as X-ray crystallograph, which usually only provide static pictures of the molecule, the current manipulation techniques, mainly including the optical tweezer, can trace the full folding/unfolding processes of single RNA by monitoring the molecule's extension or force exerted on it in real time [1,2,3].
As many nano-or mesoscopic systems, the behavior of single RNA (∼30 nm) in light tweezer is highly dynamic and noisy. The situation could become more complicated in practice: in order to manipulate single RNA by the optical trapping method, the RNA must first be tethered between two large dielectric beads (∼µm) through two long double-stranded DNA/RNA handles (∼µm); see Fig. 1. Due to the presence of the beads and handles, it would be expected that the kinetics of the RNA observed in the light tweezer experiment is distinct from the kinetics of the linker-free RNA. Hence, how to extract the intrinsic kinetic information of single RNA from experimental data is an intriguing biophysical issue. One of the possible strategies is to find optimal experimental conditions through experimental comparison and computational simulation [3,4]. Alternative way is to collect the existing RNA kinetic data and infer the intrinsic parameters by advanced statistic approaches. To the best of our knowledge, the latter was not quantitatively implemented in literature. In this Communication, we present such an effort. Physical model. Forced folding/unfolding single RNAs could be achieved in two types of manipulation experiments. One is the constant force mode (CFM), where the experimental control parameter, a constant force F of preset value, is applied on the bead in the light tweezer with or without feedback control [2,3]. The other is the passive mode (PM), where the control parameter, the distance between the centers of the light tweezer and the bead held by the micropipette, x T , is left stationary (see Fig. 1). The RNA and light tweezer system involves several time scales: the relaxation time of the bead in the tweezer, τ b , the relaxation time of the handles and single-stranded (ss) RNA, τ h and τ ssRNA , the characteristic time of the overall kinetics of the RNA, τ f-u , and the characteristic time of the opening/closing of single base pairs τ bp [4,5]. Under the conventional experimental conditions [1,2,3], the relaxation time τ h , τ ssRNA and τ bp is always far shorter than the relaxation time of the bead and overall RNA kinetics [4,5]. It is plausible to assume that the RNA is two-state, i.e., folded (f) or unfolded (u), and the extension of the handles and ssRNA is in thermal equilibrium instantaneously. Note that we do not require that the relaxation of the bead in the light tweezer is also instantaneous.
Our model involves two freedom degrees: one is the state of the RNA; the other is the distance x between the centers of the two beads. Because the force directly controlling the kinetics of the RNA is always fluctuating with time, we describe the experimental system by the following two coupled diffusion-reaction equations:
where P i (x, t) is the probability distribution of the RNA at state i (f or u) and the distance having a particular value x at time t. The Fokker-Planck operators L i in the above equations are
where D is diffusion coefficient, β -1 = k B T with k B being the Boltzmann’s constant and T the absolute temperature; V i (x) is the RNA state-dependent potential and defined as [6,7] with the persistent length P i eff [8] and contour length l i = 2L h + L i ssRNA ; and the external work W ext (x) done by the external force is F x in the CFM and ε(x T -x) 2 /2 with a tweezer stiffness ε in the PM, respectively. For the “reaction” rates k i (x), though there are significant debates about the correctness of the Bell formula, k(f [9] in describing biological molecule’s rupture or unfolding, where k 0 is the intrinsic rate constant in the absence of force, and x ‡ is the transition state location, we still use this phenomenological formula with a slight modification rather than other improved rate models having certain microscopic explanation [10,11,12,13]. Our consideration is as follows. First the Bell formula is still the simplest and most widely used in single molecule studies. Particularly, it seems to work quite well in the real RNA folding/unfodling experiments [1,2,3]. Second, other rate formulas are all model-dependent; whether they are indeed suitable to the “macroscopic” RNA folding/unfolding is not undoubted. The rate invoked here is
where k u 0 and d ‡ f are respectively the intrinsic unfolding rate in the absence of force and the transition state location away from the folded RNA state. This modification is necessary, in that the unfolding rate given by the Bell formula increases too fast with force [14]. Interestingly, it
