Quantitative Interpretation of Transverse Spin Relaxation by Translational Diffusion in Liquids Under Arbitrary Potentials

Intermolecular spin relaxation by translational motion of spin pairs have been widely used to study properties of the biomolecules in liquids. Notably, solvent paramagnetic relaxation enhancement (sPRE) arising from paramagnetic cosolutes has gained attentions for various applications, including the structural refinement of intrinsically disordered proteins, cosolute -induced protein denaturation, and the characterization of residue -specific effective ne ar-surface electrostatic potentials (ENS). Among these applications, the transverse sPRE rate known as G2 has been predominantly been interpreted empirically as being proportional to norm. In this study, we present a rigorous theoretical interpretation of G2 that it is instead proportional to norm and provide explicit formula for calculating norm without any adjustable parameters . This interpretation is independent of the type or strength of interactions and can be broadly applied, including to the precise interpretation of ENS. .CC-BY-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 22, 2024. ; https://doi.org/10.1101/2024.08.21.609078doi: bioRxiv preprint 3

Solvent paramagnetic relaxation enhancement (sPRE) has been used to study protein solvent accessibility,1-7 to refine NMR protein structures,8-10 to investigate protein-cosolute interactions,11- 14 and to characterize the electrostatic potential on small molecules and proteins.15-26 In m any sPRE applications, the transverse sPRE rate G2 is directly used to quantify electrostatics20-25 and for structure refinement ,8-10 based on the Otting-LeMaster’s empirical analysis,3, 4 which correlates G2 with the average interspin distance 〈𝑟!"〉#$%& . This empirical approach has demonstrated excellent correlation with the Poisson-Boltzmann based theoretical electrostatic near -surface potentials (ENS) and the experimental ENS. However, d espite its apparent success, a fully satisfactory theoretical explanation for why this approach works has yet to be established. In previous work, we hypothesized that G2 is proportional to 〈𝑟!'〉#$%& and demonstrated its validity using a simple hard-sphere square potential model.13 Here, we present a comprehensive theoretical framework that establishes G2 as being proportional to 〈𝑟!'〉#$%& rather than 〈𝑟!"〉#$%&. Our derivation is broadly applicable to various types of intermolecular potential, including those in confined environement, such as reverse micelle systems. While this work focuses on sPRE, the analysis is equally applicable to a wide range of intermolecular spin relaxation by translational diffusion in liquids. Theory The dipole-dipole correlation function in an isotropic liquid is given by:27, 28 𝐶(𝑡) = 𝑁! 〈"!#$̂(')∙$̂(*)+ $"(')$"(*) 〉 (1) where r and 𝑟̂ are the lengths and orientation of the interspin vector 𝑟⃗; P2(x) is the Legendre polynomial of degree 2; and NS is the number of paramagnetic cosolute molecules in the system. The number density of the cosolute 𝑛( is related to NS and the volume of the system V by ,# - . The subscript o denotes the quantity at t = 0. The spectral density J(w) is given by the cosine transform of C(t): .CC-BY-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 22, 2024. ; https://doi.org/10.1101/2024.08.21.609078doi: bioRxiv preprint 4 𝐽(𝜔) = ∫ 𝑑𝑡 ∞ 0 𝐶(𝑡) cos(𝜔𝑡) (2) where w is the angular frequency in radians.s-1, and is related to the spectrometer frequency (n) in Hz) by 𝜔 = 2𝜋𝜈. The transverse ( G2) sPREs are defined as the difference in the transverse relaxation rates, respectively, of a protein nuclear spin (generally a proton) in the presence and absence of the paramagnetic cosolute.29 At high external magnetic field limit, G2 is related to the spectral density function J(w) by:30-32 Γ) = * + - ,! '𝜋. ) ℏ) 𝛾- ) 𝛾.) 2𝐽(0) + / ' 𝐽(𝜔-)5 (3) where gH and ge are the gyromagnetic ratios of the proton and electron, respectively; µo is the vacuum permittivity constant; is Planck’s constant divided by 2 p; wH is the angular frequency of the proton. In many instances, 𝐽(0) ≫ 𝐽(𝜔-) thus Eq.(3) can be approximated as Γ) ≈ * + - ,! '𝜋. ) ℏ) 𝛾- ) 𝛾. ) 𝐽(0) (4) Consequently, the insights we can gain about the cosolute-protein interactions from the transverse sPRE relaxation rate is contained primarily in the details of the spectral densities J(0). To relate experimentally observable sPRE relaxation rates to physical quantities, we introduced the concepts of the concentration normalized average interspin distances11, 13, 14 〈𝑟!) 0〉#$%& = 4𝜋 ∫ 𝑑𝑟 ∞ 1 𝑟!) (03*)𝑒!56(7) (5) where 𝑈(𝑟) is the potential of the mean force ; b is the thermodynamic beta; and l is 2, 3, or 4. It should be emphasized that Eq.(5) is defined for any arbitrary shapes of the protein and cosolute and for arbitrary interactions between them. If J(w) can be measured across a wide range of the spectrometer fields, 〈𝑟!"〉#$%& can be calculated by taking the integration the spectral density function.11 It has also been shown that the high-frequency limit of the w2J(w) is directly related to 〈𝑟!8〉#$%&.13 ! .CC-BY-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 22, 2024. ; https://doi.org/10.1101/2024.08.21.609078doi: bioRxiv preprint 5 In the previous work,13 we hypothesized that J(0) is related to 〈𝑟!'〉#$%& by 〈𝑟!'〉#$%& ≈ )9 :"#$%& ';' 𝐽(0) (6) We demonstrated that Eq.(6) provide reasonable approximat ions for simple models involving spherically symmetric square -well and Coulombic potentials. We have also compared experimental ENS and Poisson -Boltzmann-based ENS , implemented with 〈𝑟!'〉#$%& interpretation, on ubiquitin and drkN SH3. For both systems, excellent agreements are achieved between calculated and experimental ENS similar to or slightly better than the conventionally used 〈𝑟!"〉#$%& interpretations. Despite the apparent success of the Eq.(6), its applicability for arbitrary potentials with varying interaction strengths remained unclear. In this work, we establish a rigorous theoretical foundation of the validity of Eq.(6) and demonstrate that it applies across a wide range of systems, including those in confined environments. In this work, we consider the simplest case where the protein and cosolutes are modeled as hardspheres, with the nuclear and electron spins located at their respective centers. The interspin vector 𝑟⃗ is specified by three coordinates in spherical coordinates 𝑟⃗ = (𝑅, 𝜃, 𝜙). The cosolute and protein are assumed to be diffuse under the influence of the spherically symmetric potential characterized by the potential of the mean force U(R), which only depends on the center-to-center distance R. The cosolute is allowed to diffuse within the volume V specified by the contact distance 𝑅< and the outer boundary 𝑅= such that 𝑅< ≤ |𝑟⃗| ≤ 𝑅=. To simplify and clarify our discussion, many of the technical details of the derivations of the equations described below are provided in the Supporting Information. Under the assumption that cosolute-protein pair is described by the Smoluchowski formalism (see Supporting Information for detail), it can be shown13 that the J(0) is given by 𝐽(0) = ('>)( + 𝑛( ∫ 𝑑𝑅? @) @* .+,-(/) @! 𝜑?(𝑅?) (7) where 𝜑?(𝑅?) is the solution to the equation, − A A@! F𝑒!56(@!)𝑅? ) A A@! 𝜑?(𝑅?)G + 6𝑒!56(@!)𝜑?(𝑅?) = +.+,-(/!) '>:"#$%&@! (8) .CC-BY-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 22, 2024. ; https://doi.org/10.1101/2024.08.21.609078doi: bioRxiv preprint 6 subjected to the reflective boundary conditions AB! A@! I @!C@* = AB! A@! I @!C@) = 0 (9) Therefore, J(0) can be calculated once 𝜑? is known. In practice, 𝜑? can be solved analytically only for few special cases such as force-free hardsphere (FFHS) model where U(R)=0 or when the potential of the mean force is modelled as a square-well potential. In the case of the FFHS model, J(0) is given by 𝐽(0) = 𝑛( >(*!D)E/) 3D(*3D)F/93D(+3/) D)GH +'@*(*3D3D( 3D13D2) (10) where 𝑦 = 𝑅;' )9 :"#$%&@* = ';' )9 :"#$%& 〈𝑟!'〉#$%& (11) It should be pointed out that for the FFHS model with infinite volume, Eq.(11) implies that Eq.(6) is exact; however, as seen from Eq.(10), 𝐽(0) is not strictly equal to ';' )9 :"#$%& 〈𝑟!'〉#$%& when finite volume is considered. Lower and Upper Bounds of J(0) Since the analytical form of 𝜑? for an arbitrary potential is not known, we approximate J(0) by evaluating its lower and upper bounds using the complementary variational principle.33-36 For rigorous explanations the technique employed in this work, see refer to Ref.33, 37, 38. Let us introduce new functions 𝑝, 𝑤, 𝑞 given by 𝑝(𝑅?) = 𝑒!56(@!)𝑅? ) 𝑤(𝑅?) = 6𝑒!56(@!) 𝑞(𝑅?) = 5𝑒!56(@!) 4𝜋𝐷J%K#L𝑅? (12) It is important to note that 𝑝, 𝑤, 𝑞 are nonnegative functions. .CC-BY-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 22, 2024. ; https://doi.org/10.1101/2024.08.21.609078doi: bioRxiv preprint 7 Using these new functions, Eq.(8) can be written in the form of Sturm-Liouville equation − A A@! -𝑝 AB! A@! . + 𝑤𝜑? = 𝑞 (13) Let us introduce a functional 𝐼[𝜑] given by 𝐼[𝜑] = ∫ 𝑑𝑅? @) @* Y−𝑝 - AB A@! . ) − 𝑤𝜑) + 2𝑞𝜑[ (14) where the admissible functions of this functional are the set of all functions 𝜑 that have well defined first- and second derivatives and also satisfy the boundary conditions Eq.(9). When 𝜑 = 𝜑?, 𝐼[𝜑?] = \ 𝑑𝑅? @) @* ]−𝑝 2𝑑𝜑? 𝑑𝑅? 5 ) − 𝑤𝜑? ) + 2𝑞𝜑?^ = \ 𝑑𝑅? @) @* {𝑞𝜑?} = 25 (4𝜋)/ 𝐽(0) 𝐷J%K#L𝑛( (15) If 𝜑 is a function that deviates from the 𝜑? by a small amount 𝜑(𝑅?) = 𝜑?(𝑅?) + 𝜀𝜂(𝑅?) (16) where 𝜀 is some small number and 𝜂 is some arbitrary admissible function. Substituting Eq.(16) in Eq.(14), we get 𝐼[𝜑] = \ 𝑑𝑅? @) @* ]−𝑝 2 𝑑𝜑 𝑑𝑅? 5 ) − 𝑤𝜑) + 2𝑞𝜑^ = \ 𝑑𝑅? @) @* ]−𝑝 2𝑑𝜑? + 𝜀𝜂 𝑑𝑅? 5 ) − 𝑤(𝜑? + 𝜀𝜂)) + 2𝑞(𝜑? + 𝜀𝜂)^ = 𝐼[𝜑?]− 𝜀) \ 𝑑𝑅? @) @* ]𝑝 2 𝑑𝜂 𝑑𝑅? 5 ) + 𝑤𝜂) ^ (17) The first variation 𝛿𝐼 (the term linear to 𝜀) is given by 𝛿𝐼 = 0 (18) and the second variation 𝛿) 𝐼 (the term linear to 𝜀) ) is given by .CC-BY-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 22, 2024. ; https://doi.org/10.1101/2024.08.21.609078doi: bioRxiv preprint 8 𝛿) 𝐼 = −𝜀) ∫ 𝑑𝑅? @) @* Y𝑝 - AM A@! . ) + 𝑤𝜂) [ ≤ 0 (19) Since 𝐼[𝜑?] has the first variation zero and second variation negative (concave down), Eq.(18) and (19) imply that 𝐼[𝜑?] is the global maximum value that can attained by any admissible 𝜑.33 To obtain the upper bound, define another function 𝑢?(𝑅?) = 𝑝(𝑅?) A A@! 𝜑?(𝑅?) (20) Note that 𝑢? satisfies 𝑢?(𝑅<) = 𝑢?(𝑅=) = 0. Then the Eq.(13) can be rewritten as − AN! A@! + 𝑤𝜑? = 𝑞 (21) Let us introduce another functional 𝐺[𝑢] given by 𝐺[𝑢] = ∫ 𝑑𝑅? @) @* Y * O 𝜇) + * P - A, A@! + 𝑞. ) [ (22) where the admissible functions is the set of all functions that have well-defined first derivative and satisfies the boundary conditions 𝑢(𝑅)1 R(1) :"#$%&;' (24) Similar to the lower bound case, we calculate the first- and second variations of 𝐺[𝑢] by considering small deviations to 𝑢?. Consider the function 𝑢 given by 𝑢(𝑅?) = 𝑢?(𝑅?) + 𝜀𝜂S(𝑅?) (25) where 𝜂S is some admissible function. When Eq.(25) is substituted in Eq.(22), we get .CC-BY-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 22, 2024. ; https://doi.org/10.1101/2024.08.21.609078doi: bioRxiv preprint 9 𝐺[𝑢] = \ 𝑑𝑅? @) @* ]1 𝑝 𝑢) + 1 𝑤 2 𝑑𝑢 𝑑𝑅? + 𝑞5 ) ^ = \ 𝑑𝑅? @) @* h1 𝑝 (𝑢? + 𝜀𝜂S)) + 1 𝑤 F𝑑𝑢? + 𝜀𝜂S 𝑑𝑅? + 𝑞G ) i = 𝐺[𝑢?]+ 𝜀) \ 𝑑𝑅? @) @* h𝑝 F 𝑑𝜂S 𝑑𝑅? G ) + 𝑤𝜂S) i (26) Eq.(26) implies that 𝛿𝐺 = 0 (27) 𝛿) 𝐺 = 2 ∫ 𝑑𝑅? @) @* Y𝑝 - AM3 A@! . ) + 𝑤𝜂S) [ ≥ 0 (28) Having first variation zero and second-variation nonnegative (concave up), Eqs.(27) and (28) implies that 𝐺[𝑢?] is the global minimum value attainable by any admissible function.33 To summarize our finding, we derived the following relation 𝐼[𝜑] ≤ 𝐼[𝜑?] = )+ ('>)1 R(1) :"#$%&;' = 𝐺[𝑢?] ≤ 𝐺[𝑢] (29) Eq.(29) provides the useful estimation of J(0) by choosing proper choice of the trial functions 𝜑 and u. Here, the FFHS model (U(R)=0) was used as the trial function by setting 𝜑 = 𝜑T--( to obtain the lower bound. Similarly, upper bound were found by substituting trivial function u = 0 into Eq.(22). After some rearrangement of the equation, we obtained the main finding of this work 0.9375〈𝑟!'〉#$%& ≤ )9 :"#$%&R(1) ';' ≤ 1.125〈𝑟!'〉#$%& (30) Eq.(30) suggest that )9 :"#$%&R(1) ';' ≈ 〈𝑟!'〉#$%& with the error up to 12.5 %. Consequently, once 𝐷J%K#L is acquired, a reasonable estimate of 〈𝑟!'〉#$%& can be determined without any adjustable parameters. It should be emphasized that Eq.(30) holds true for arbitrary potential of any interaction strength. Furthermore, Eq.(30) does not depend on the value of 𝑅= (see Supporting Information and Fig.S1), making it applicable to confined environments, such as reverse- micelles, under assumption that the diffusive motion of the cosolute is described by .CC-BY-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 22, 2024. ; https://doi.org/10.1101/2024.08.21.609078doi: bioRxiv preprint 10 Smoluchowski equation Eq.(S2). As a result, Eq.(30) provides the rigorous interpretation of J(0) for wide variety of systems. To illustrate this, we simulated the spectral densities using Lennard- Jones and Coulombic potentials and demonstrated that the inequality Eq.(30) indeed holds valid (Figure 1). Interestingly, the upper bound found in Eq.(30) combined with the continuity of the first derivative of 𝜑? implies that AB! A@! is always less than or equal to zero. That is, 𝜑?(𝑅?) is a monotonically decreasing function, which may not be intuitively obvious.

We have developed a new quantitative interpretation of J(0) for intermolecular spin relaxation, directly related to the transverse relaxation rate by considering the lower and upper bounds of J(0) in terms of 〈𝑟!'〉#$%&. Specifically, Eq.(30) demonstrates that J(0) is proportional to 〈𝑟!'〉#$%&, in contrast to the widely used assumption that J(0) is proportional to 〈𝑟!"〉#$%&. Since Eq.(30) holds true for arbitrary potential, including electrostatics and van der Waals interactions, our finding provides a rigorous basis for interpreting sPRE applications on protein structure refinement and the calculation of ENS. Importantly, our approach to calculating 〈𝑟!'〉#$%& is absolute without involving any adjustable parameters. In this study, we focused on a spherically symmetric potential. Future work will investigate the validity of Eq.(30) in more complex potentials with more intricate protein and cosolute shapes.

This work was funded by the start-up funds by Washington University in St. Louis. .CC-BY-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 22, 2024. ; https://doi.org/10.1101/2024.08.21.609078doi: bioRxiv preprint 11

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Calculus of variations; Courier Corporation, 2000. .CC-BY-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 22, 2024. ; https://doi.org/10.1101/2024.08.21.609078doi: bioRxiv preprint 13 Figures Figure 1. Illustration of the validity of Eq.(30). (A) The examples of the potentials used to simulate 〈𝑟45〉6789 and the approximate 〈𝑟45〉6789 :;;87< = 27𝐷trans𝐽(0) 4𝑛S . The Lennard-Jones potential of the form 𝛽𝑈(𝑅C) = 4𝑎 , - D! D" . EF − - D! D" . G 0 was used to generates the points on the left panel. The potential is minimum at 2 # $𝑅H. The Coulomb potential of the form 𝛽𝑈(𝑅C) = −𝑎 D%I&'(" D" where 𝜅 is the inverse of the Debye length was used to simulate the points on the right panel. For both potentials, 𝑎 is the parameter used to modulate the strengths of protein-cosolute interaction. (B) The comparison between exact and approximate 〈𝑟45〉6789. The exact 〈𝑟45〉6789 were calculated from numerically evaluating the integral Eq.(5). The approximate 〈𝑟45〉6789 were calculated using Eq.(6). The corresponding 〈𝑟45〉6789 :;;87< are numerically calculated by finite-difference method explained in details in the Supporting Information of the Ref. 13. The relevant parameters used for the finite-difference methods are : ∆𝑅 = 104EF m, 𝑀 = 120, 𝑁 = 500 and 𝑐 = 1.01. The values for the other parameters used in the simulations are: 𝑅J = 1 nm, 𝑅K = 5.8 nm, 𝑅H = 1.15 nm and the Debye lengths 1/𝜅 = 25 nm. It should be pointed out that the numerical values of 𝑛( and 𝐷L8:6M do not affect our results. .CC-BY-ND 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted August 22, 2024. ; https://doi.org/10.1101/2024.08.21.609078doi: bioRxiv preprint