Deconvoluted extension probability analysis

Hyun-Kyu Choi; Hyunook Kang; Chanwoo Lee; Hyun Gyu Kim; Ben P. Phillips; Soohyung Park; Charlotte Tumescheit; Sang Ah Kim; Hansol Lee; Soung-Hun Roh; Heedeok Hong; Martin Steinegger; Wonpil Im; Elizabeth A. Miller; Hee-Jung Choi; Tae-Young Yoon

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Deconvoluted extension probability analysis

HC Hyun-Kyu Choi

HK Hyunook Kang

CL Chanwoo Lee

HK Hyun Gyu Kim

BP Ben P. Phillips

SP Soohyung Park

CT Charlotte Tumescheit

SK Sang Ah Kim

HL Hansol Lee

SR Soung-Hun Roh

HH Heedeok Hong

MS Martin Steinegger

WI Wonpil Im

EM Elizabeth A. Miller

HC Hee-Jung Choi

TY Tae-Young Yoon

This method is extracted from research article: Nat Chem Biol, Apr 2022

Evolutionary balance between foldability and functionality of a glucose transporter

DOI: 10.1038/s41589-022-01002-w

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To obtain an extension distribution of single GLUT3 protein $({\tilde{P}}_{p} (Z_{p}; F))$ with Brownian noises of magnetic beads and handles removed, we implemented deconvolution of the measured marginal probability distribution $({\tilde{P}}_{m} (Z_{m}; F))$ in real space as previously established in optical tweezers studies^69,70. Because the magnetic bead in magnetic tweezers is not physically trapped unlike with optical tweezers (i.e., magnetic force is not a fluctuating variable but stably fixed), the marginal probability distribution $({\tilde{P}}_{m} (Z_{m}; F))$ from Hamiltonian of the bead in the presence of magnetic force could be directly described as ${\tilde{P}}_{m} (z_{m}; F) \approx \frac{1}{4} e^{β F z} P_{m} (z) = \frac{1}{4} P_{m} (z; F)$ where P_m(z; F) is the measured equilibrium probability of the total bead-handle-protein system with separation z at the constant force 𝓕; β is 1/k_BT. By performing deconvolution in real-space, we can derive the following integral.

Where ${\tilde{P}}_{bh} (z; F)$ is conjugated probability of handles (PEG polymers (peg), two DNA handles (dh; dh1 defined as DNA handle directing towards magnetic bead, dh2 towards peg) and two polypeptide linkers (ph) between DNA and GLUT3) and magnetic bead. In brief, ${\tilde{P}}_{bh} (Z; F) \equiv ℱ^{- 1} ({\tilde{P}}_{b} (k; F) {\tilde{P}}_{peg} (k; F) {\tilde{P}}_{dh 1} (k; F) {\tilde{P}}_{dh 2} (k; F) {\tilde{P}}_{ph}^{2} (k; F))$ where F⁻¹ indicates inverse Fourier-transformation and k is the wave-vector in Fourier-space. The probability of the magnetic bead, ${\tilde{P}}_{b} (k; F) is \frac{f \sinh ((f - i k) R_{b})}{(f - i k) \sinh (f R_{b})}$ where R_b is the radius of the magnetic bead f, βF and i is the complex number. The rest terms in ${\tilde{P}}_{bh} (Z; F)$ can be described by ${\tilde{P}}_{j} (k; F) \equiv \frac{\sum_{n, l, l'} Ψ_{l^{'}, B . C}^{T} (L_{c, j}) {[e^{- E_{n, j} (f - i k) L_{C, j}}]}_{l^{'}, l} Ψ_{l, B . C} (0)}{\sum_{n, l, l'} Ψ_{l^{'}, B . C}^{T} (L_{c, j}) {[e^{- E_{n, j} (f) L_{c, j}}]}_{l^{'}, l} Ψ_{l, B . C} (0)}$ where the index j represents the components composed of peg, dh1, dh2 and ph. The corresponding total contour length is L_c,j. Ψ_B.C and E_n,j are an eigen state and eigen value (total energy), respectively as previously defined and estimated from effective Hamiltonian equation of propagator of biopolymer in Markovian regime⁶⁶. Index B.C in eigen state indicates whether semi-flexible biopolymer is half-constrained (one side of peg and dh1) or unconstrained (dh2, ph).

To avoid any numerical instability and ill-conditioned result, we used suitable fitting functions for all probability distributions ( ${\tilde{P}}_{bh}, {\tilde{P}}_{p}$ and ${\tilde{P}}_{m}$ ). Linear superposition of Gaussians was employed to determine the pure probability of GLUT3 (we used median-filtered traces with 5-Hz window. Because the characteristic time scale of magnetic bead is less than 30 ms, we could implement deconvolution of the behavior of the bead from the measurement).

Where λ means bh (handles and bead), p (GLUT3) or m (total system) and $g (z_{λ}, μ_{i}^{λ}, σ_{i}^{λ})$ is Gaussian distribution ( ${(2 π σ_{i}^{λ^{2}})}^{- 1 / 2} e^{- {(z_{λ} - μ_{i}^{λ})}^{2} / 2 σ_{i}^{λ^{2}}}$ ). $w_{i}^{λ}$ is weighting factor in linear combination and N_λ is total number of Gaussian components (for simplicity, N_bh = 1 was chosen). Then, parameters of the deconvoluted extension distribution of the single GLUT3 are described as $w_{i}^{p} \approx w_{i}^{m}, μ_{i}^{p} \approx μ_{i}^{m} - \sum_{j = 1}^{N_{bh}} w_{j}^{bh} μ_{j}^{bh}$ and $σ_{i}^{p^{2}} = σ_{i}^{m^{2}} - σ_{i}^{{bh}^{2}} - 2 c o v (z_{m}, z_{bh})$ . For ensemble averages of the deconvoluted probability distributions, weighted arithmetic mean was used to visualize the average probability distribution (i.e., $〈 {\tilde{P}}_{p} (z_{p}; F) 〉 \equiv \sum_{m = 1}^{M} a_{m} {\tilde{P}}_{p, m} (z_{p}; F) / \sum_{m = 1}^{M} a_{m}$ where m is the number of traces, M is the total number of measured traces and a_m is the normalized weighting factor, which depends on sample size in each trace.

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