September 25, 2017, Monday
University of Colorado at Boulder Search A to Z Campus Map University of Colorado at Boulder CU 
Search Links


MBW:Approximating free energy profiles from single-molecule measurements

From MathBio

Jump to: navigation, search

Summary

In the field of single molecule biophysics, various questions remain elusive. The most notable is the question of protein folding: How does a linear sequence of amino acids fold into a functional three dimensional structure? In general, the folding process is viewed as movement over a free energy landscape. Under Brownian motion, the protein samples the energy landscape until it becomes trapped in an energy well corresponding to the folded state. Recently developed techniques allow experimentalists to apply piconewton scale forces to single molecules while measuring nanometer scale extensions, probing the energy landscape of the system. However, in such experiments, the molecular free energy landscape is perturbed by the measurement device, complicating analysis. Hummer and Szabo have developed a method to reconstruct free energy landscapes from experimental data.[1] Free energy differences can be written as a function of the free energy landscape. The technique is derived by transforming this relationship into a form that is more readily approximated. Based on simulation results, the new reconstruction scheme performs significantly better than previous approximate energy landscape reconstruction techniques.

Background

Single Molecule Assays

The experimental assay for single molecule force extension experiments consists of the molecule of interest fixed at one end and attached to a force actuator at the other. The force actuator is typically a polystyrene bead held in an optical trap or the microscopic cantilever of an atomic force microscope (AFM). Generally, the actuator is Hookean. In this case, the system can be modeled as the molecule and a harmonic spring in series attached to surfaces at either end. Then, the total distance can be controlled while observing molecular extension.

Relation to Energy Profiles

The projection of a multidimensional energy landscape onto a single reaction coordinate is known as an energy profile. In force extension experiments, the most intuitive reaction coordinate is molecular extension. Boltzmann’s law states that, in equilibrium, the probability of finding a system in a given state corresponds with the energy of the state. Ideally, measuring the state distribution of a system in equilibrium is equivalent to measuring the energy landscape. In the case of single molecule experiments that are not at constant force, this is not the case. The system is not in equilibrium and the free energy landscape is distorted by the time-varying applied force. Fortunately, Jarzynski's equality, relating differences in free energy at equilibrium to irreversible work, can be used to estimate the equilibrium free energy. The separation of the molecular free energy landscape from that of the system is the focus of the mathematical methods discussed below.

Nucleic Acid Hairpins

Schematic of a RNA hairpin. (image from http://en.wikipedia.org/wiki/Stem-loop)

Due to the complexity of the energy landscapes for protein folding, the model molecule for studying energy profiles is a nucleic acid hairpin. Nucleic acids are polymers of a limited alphabet of nucleotides. In the case of DNA, the nucleotides are adenine (A), thymine (T), guanine (G), and cytosine (C), where A binds specifically with T and G with C. A single strand of DNA can be constructed in such a way that it folds upon itself, creating a structure called a hairpin or stem-loop (right). In the simplest case, a nucleic acid hairpin is bistable--either open or closed. The corresponding energy profile would have two wells corresponding which either stable state separated by a barrier. Although much simpler than the energy landscape for a protein, useful properties, such as transition rates between the two states, can be derived from the hairpin energy profile using Kramers' theory of diffusion over a barrier.

History

The fundamental theory for understanding molecular transitions between different states in terms of energy landscapes was initiated 1940 with the inception of Kramer's theory.[2] However, due to the recent discovery of Jarzynski's inequality in 1997 and development of single molecule manipulation techniques, all work for approximating free energy profiles from experiment has occurred in the past decade. Nevertheless, much progress has been made in developing methods to estimate free energy profiles from single-molecule experiments. The original method, also developed by Hummer and Szabo, involved a more direct application of Jarzynski's equality.[3]. The method discussed in that paper constructs the free energy profile by a weighted historgram by work. Since, a method known as the stiff-spring approximation was developed. [4] The stiff-spring approximation represents the free energy landscape in terms of a second order approximation of Jarzynski's equality. Because of the low order of approximation, it is only valid when the spring stiffness is significantly higher than the maximum curvature in the free energy profile. The work discussed in this article is the third method for free energy profile reconstruction.

Mathematical Model

Energy Profile Approximation

Consider a single molecule force extension experiment where the system extension is prescribed by z(t). Denote q(x) as the molecular extension of state x and let k be the spring constant of the force actuator. We will construct an approximation for the molecular free energy landscape, G0(q), as done by Hummer and Szabo.[1] Begin with the Hamiltonian, H, of the entire system:

H[x,z(t)]=H_{0}(x)+V[x,z(t)]

where H0(x) is the Hamiltonian of the molecular system and V[x, z(t)] is the contribution from the experimental assay, namely:

V[x,z(t)]=k(q(x)-z(t))^{2}/2

Let A(z) denote the free energy difference between positionsz(t) and z(0), which is by definition:

e^{{-\beta A(t)}}={\frac  {\int dxe^{{-\beta H[x,z(t)]}}}{\int dxe^{{-\beta H[x,z(0)]}}}}

where β-1=kBT with kB as the Boltzmann constant and T as the temperature. Using Jarzynski's equality, this equilibrium free energy difference can be written in terms of nonequilibrium work as an average over all trajectories starting from H[x, z(0)] at equilibrium

e^{{-\beta A(t)}}=\langle e^{{-\beta W(z)}}\rangle

where work, W(z), is defined as

W(z)=\int _{0}^{t}{\frac  {\partial V}{\partial z}}{\frac  {dz}{dt'}}dt'

With some manipulation, we can write the original expression for A(z) in terms of G0 as

e^{{-\beta A(z)}}=(2\pi /\beta k)^{{1/2}}e^{{frac{1}{2\beta k}{\frac  {d^{2}}{dx^{2}}}}}e^{{-\beta G_{0}(z)}}

or, solved for G0, as

e^{{-\beta G_{0}(z)}}=(2\pi \epsilon )^{{-1/2}}e^{{{\frac  {\epsilon d^{2}}{2dx^{2}}}}}e^{{-\beta A(z)}}

where ε=(βk)-1.

Next, this exponential can be rewritten using the Hubbard-Stratonovich transform, under which

e^{{{\frac  {\epsilon d^{2}}{2dx^{2}}}}}=(2\pi \epsilon )^{{-1/2}}\int d\zeta e^{{-\zeta ^{2}/2\epsilon }}e^{{i\zeta {\frac  {d}{dz}}}}

so

e^{{-\beta G_{0}(z)}}=(2\pi \epsilon )^{{-1}}\int d\zeta e^{{-\zeta ^{2}/2\epsilon -\beta A(z+i\zeta )}}


To approximate the integral, we can set ζ0 so that

{\frac  {\partial [-\zeta ^{2}/2\epsilon -\beta A(z+i\zeta )]}{\partial \zeta }}(\zeta =\zeta _{0})=0

which is

\zeta _{0}=-i\epsilon \beta A(z+i\zeta _{0})

This yields, after integration,

e^{{-\beta G_{0}(z)}}\approx (2\pi \epsilon (1-\epsilon \beta {\ddot  {A}}))^{{-1/2}}e^{{-\beta A+\epsilon \beta ^{2}{\dot  {A}}^{2}/2}}

Finally, taking the logarithm and neglecting the constant term, we get

G_{0}\left(q=z-{\frac  {{\dot  {A}}(z)}{k}}\right)\approx A(z)-{\frac  {{\dot  {A}}(z)^{2}}{2k}}+{\frac  {1}{2\beta }}ln\left(1-{\frac  {{\ddot  {A}}(z)}{k}}\right)

Here we have derived an approximation for the free energy profile as a function of derivatives of the free energy differences, which can be easily calculate using Jarzynski's equality as stated above.

In the presence of linkers

Many single molecule experimental assays are more complex than solely a spring attached to the molecule of interest. Instead, the molecule is usually attached to both the spring and the surface via molecular linkers. For example, in the case of a DNA hairpin, the single stranded DNA of the hairpin must have double stranded DNA handles in order for it to be long enough to be studied with optical tweezers. The double stranded DNA handles are elastic. Hummer and Szabo attempt to incorporate the linkers into their reconstruction method and roughly approximated the free energy surface when the linkers are sufficiently soft that the information seen from the molecule is whether it's in an open or closed state. This discussion is restricted to a bistable nucleic acid hairpin. Although not generally true, the linkers are assumed to be harmonic. Additionally, Hummer and Szabo introduce a second reaction coordinate, Q, into the approximation of the free energy surface in order to more accurately calculate rates. The free energy surface of the system becomes

G=G_{M}(x,Q)+{\frac  {1}{2}}k_{L}(x-q)^{2}+{\frac  {1}{2}}k[q-z(t)]^{2}

where x is the molecule's extension, GM is the free energy surface of the molecule, kL is the spring constant of the linker, q is the extension of the molecule and the linkers, and z(t) is the extension of the entire system. Let DQ, 'Dx, and 'Dq be diffusion coefficients associated with Q, x, and q. Then the dynamics of the system can be written in terms of reaction-diffusion equations for the transition probabilities p1(q, t) and p2(q, t) for the closed and open states.

{\frac  {\partial p_{1}}{\partial t}}=-k_{1}(q)p_{1}+k_{2}(q)p_{2}+D_{q}{\frac  {\partial }{\partial q}}e^{{-\beta V_{1}}}{\frac  {\partial }{\partial q}}e^{{\beta V_{1}}}p_{1}

{\frac  {\partial p_{2}}{\partial t}}=k_{1}(q)p_{1}-k_{2}(q)p_{2}+D_{q}{\frac  {\partial }{\partial q}}e^{{-\beta V_{2}}}{\frac  {\partial }{\partial q}}e^{{\beta V_{2}}}p_{1}

where Vi(q, t) is the potential mean force along q in state i, which is approximately

V_{i}(q,t)={\frac  {1}{2}}k_{L}(x_{i}-q)^{2}+{\frac  {1}{2}}k(q-z(t))^{2}+G_{i}^{0}.

where Gi0 is the free energy of state i and xi is the extension of state i. The free energy landscape of the molecule/linker system can be derived from the above approximation. Then, using the diffusion equations, the transition rates can be approximated using two dimension Kramers' theory.

Analysis and Interpretation

Viability of energy landscape approximation

In order to demonstrate the utility of the derived approximation technique, Hummer and Szabo simulated various A(z) curves and reconstructed the energy profiles (top right). The error in the approximation is related to the stiffness, k of the spring. In the figure, the exact energy landscape is shown by red squares. The reconstructed landscape using the method developed here, known as the quasi-harmonic approximation, is shown by the green dashed line. The previous best approximation, known as the stiff spring approximation, is shown by the blue solid line. From top to bottom in the figure, the spring stiffness is decreasing. Qualitatively, the relationship between the maximum curvature of the energy landscape and the curvature corresponding to the harmonic well of the spring is useful. As the curvature of the spring becomes smaller than the maximum curvature of the energy profile, the stiff spring approximation becomes worse rapidly while the quasi-harmonic approximation continues to be a reasonable model. As expected, the new approximation technique is significantly better at reconstructing the energy profile as the stiffness decreases.

Energy landscape approximation in presence of linkers

To asses the performance of the discussed system with linkers, Hummer and Szabo conducted more simulations (bottom right). In this case, they simulated a RNA hairpin with soft linkers at various pulling speeds. They then attempted to reconstruct the energy landscape of the hairpin and linker system. Figure A shows the landscape of the folded, closed state G1 (dotted green) and the unfolded, open state G2 (dotten blue). Inset is a sample simulated data trace. Figure B shows a reconstructed free energy landscape of the molecule and linker system. The various symbols correspond to profile reconstructions at different force loading rates while the red line is the exact free energy landscape. Even in the presence of linkers, the approximation method sufficiently approximates many features of the true energy landscape.

References

  1. 1.0 1.1 Hummer, G. and A. Szabo (2010). "Free energy profiles from single-molecule pulling experiments." Proc Natl Acad Sci U S A 107(50): 21441-21446
  2. Kramers, H.A. (1940) Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7(4):284-304
  3. Hummer, G. and A. Szabo (2001). "Free energy reconstruction from nonequilibrium single-molecule pulling experiments." Proc Natl Acad Sci U S A 98(7): 3658-3661
  4. Park, S, Khalili-Araghi F, Tajkhorshid E, Schulten K (2003) Free energy calculation from steered molecular dynamics simulations using Jarzynski's equality. J Chem Phys 11:3559-3566.