Thermal conductivity and conductance of protein in aqueous solution: Effects of geometrical shape

Abstract Considering the importance of elucidating the heat transfer in living cells, we evaluated the thermal conductivity κ and conductance G of hydrated protein through all‐atom non‐equilibrium molecular dynamics simulation. Extending the computational scheme developed in earlier studies for spherical protein to cylindrical one under the periodic boundary condition, we enabled the theoretical analysis of anisotropic thermal conduction and also discussed the effects of protein size correction on the calculated results. While the present results for myoglobin and green fluorescent protein (GFP) by the spherical model were in fair agreement with previous computational and experimental results, we found that the evaluations for κ and G by the cylindrical model, in particular, those for the longitudinal direction of GFP, were enhanced substantially, but still keeping a consistency with experimental data. We also studied the influence by salt addition of physiological concentration, finding insignificant alteration of thermal conduction of protein in the present case.


SI-1 Gyration-radius-like geometry measure for uniform sphere and cylinder
and colleagues 1 used a radius of gyration for a measure of spherical protein size, whereas their idea to describe protein size is not unique. Here, we consider an ideal sphere or cylinder with uniform mass and compare the radius of gyration (or gyration radius-like geometry measure) with the given geometry measure.
First, we discuss a sphere of uniform mass with radius of Rs, where the center of mass is located at the origin. The mass density (total mass is taken to be unity) and volume are given as follows: Then, we can analytically calculate the radius of gyration, , : , 2 = � � � ⋅ ( 2 + 2 + 2 ) ( , , ) We can thus find that the given radius, Rs, is � 5 3 ≈ 1.29-fold larger than the radius of gyration. 1.29 can be a scale factor to correct the value of , for the actual radius value. Recalling eqn. 8 in the main text, using , for estimation of κ leads to the overestimation by 1.29.
Second, we consider a cylinder of uniform mass with radius of Rc and height of Hc, where the center of mass is located at the origin. The mass density and volume are given as follows: Then, we can analytically calculate the radii of gyration extended for the cylindrical radius and height, denoted as , and ℎ , : Accordingly, correction factors to recover the given geometry measures are c.a. √2 ≈ 1.41 and √3 ≈ 1.73 for the radius and height, respectively. Furthermore, we can calculate an effective gyration radius of the cylinder regarded as a sphere, | ; the formula is given as: According to eqn. 8 in the main text, the thermal conductivity is proportional to 2 , so that a given length measure l and a calculated protein volume V are important factors to determine the value of thermal conductivity, κ. It is thus worthwhile comparing the values of κ between different geometry definitions.
First, we consider the effect of correction of geometrical measure (tentatively given by the gyration radius) for the sphere with radius of Rs. As shown above, , is c.a.
0.77-fold smaller than Rs. Using , as an effective radius, the volume becomes less than half of the exact volume of the sphere as shown below: This gives two cautions to use the radius of gyration to describe the shapes of proteins.
One is that the value of κ is overestimated by 1.29-folds. The other is that the boundary between a protein and the solution environment is not correctly described due to the smaller radius value defined as the radius of gyration. From theoretical point of view, it seems to be reasonable to numerically correct , , which is calculated from the atomic coordinates of protein, by multiplying the factor of 1/0.77 = 1.29.
Second, we consider correction effects for the cylinder with radius of Rc and height of Hc. As addressed above, , and ℎ , are 0.71-fold and 0.58-fold smaller than Rc and Hc, respectively. As for the volume calculated by using these gyration-like measures, we find Thus, using the radius and height of gyration-like geometry measure leads to c.a. As illustrated in Figure S1, this ratio is greater than unity, indicating that the sphere model overestimates the volume of protein with cylindrical shape.

SI-2 Calculation of geometry measure (radius and height) for cylinderlike body by using atomic coordinates of protein molecule
We calculate the radius and height values for the two model proteins, myoglobin (Mb) and green fluorescent protein (GFP), described in terms of the cylindrical model as follows. First, we choose two Cα carbon atoms, which appear to be relatively stable in the protein (such as those located in an α-helix), and define a unit vector, �⃗ =   Figure 1 in the main text.

SI-3 t-r dependent solution to thermal diffusion equation in the cylindrical model
We solve the thermal diffusion equation for the cylindrical model by ignoring angle (θ) and height (z) dependence. The differential equation is given below Using the following transformation, we can make eqn. 15 non-dimensional form, that is, eqn. 17 below. It is noted that l can be selected arbitrarily: If we consider a variable-separated solution, Eqn. 17 becomes where λ denotes a real positive number. As for the function T, the solution is given as Meanwhile, function X satisfies the first-class Bessel differential equation: Thus, we can give the solution to eqn. 17 as It is noted that 0 (•) is the 0 th order first-class Bessel function. Using eqn. 22, we can construct a general solution: Here, we suppose that the cylinder has radius of Rc and height of Hc.
The right-hand side of eqn. 23 is multiplied by − and is substituted into B.C.: Then, by combining the eqns. 26 and 27, we finally obtain By substituting eqn. 23 into I.C., we obtain the following equation: Eqn. 29 is multiplied by 0 � � and integrated for r from 0 to Rc: On the right-hand side, if index m is not equal to index n, terms in the summation vanish due to the relation: This relation can be obtained by combining the formula of the first-class Bessel function and eqn. 28, which can be confirmed easily. 1 (•) is the 1 st order first-class Bessel function. According to the eqns. 30 and 31, the coefficient has the following form: Here, we use Rc as l in eqn. 32 and substitute the into eqn. 23, leading to By substituting this equation into the thermal diffusion equation dependent on time and radius, eqn. 15, we can obtain the relation between thermal diffusion coefficient and geometrical measure: As in the case of the earlier study by Lervik, 1 we averaged eqn. 33 over the protein volume: It is noted that is 2 . This is the formula for simulation data fitting, referred to as eqn. 3a in the main text, by retaining only the first term (n = 1) in the n series which is expected to decay rapidly as n increases.

SI-4 t-z dependent solution to thermal diffusion equation in the cylindrical model
We solve the thermal diffusion equation for the cylindrical model by ignoring angle (θ) and radius (r) dependence. The differential equation is given below: Using the following transformation, we can make eqn. 36 a non-dimensional form, that is, eqn. 38 below. It is noted that l can be selected arbitrarily: where λ denotes a real positive number. As for the function T, the solution is given as Meanwhile, function X satisfies It is noted that can be set to 0 due to the inversion symmetry of cylinder around the origin so that we use the following form hereafter: Using eqn. 45, we can construct a general solution: The right-hand side of eqn. 46 is multiplied by − and is substituted into B.C.: Then, by combining eqns. 49 and 50, we finally obtain By substituting eqn. 46 into I.C., we obtain the following equation: Eqn. 52 is multiplied by and integrated for z from 0 to HC/2, then leading to On the right-hand side, if index m was not equal to index n, terms in the summation vanished due to the combination of eqn. 51 with the following relation, According to eqn. 53, the coefficient has the following form: Here, we use Hc/2 as l in eqn. 55 and substitute the into eqn. 46, then leading to By substituting this equation into the thermal diffusion equation dependent on time and height, eqn. 36, we can obtain the relation between thermal diffusion coefficient and geometrical measure: As in the case of the earlier study by Lervik, 1 we averaged eqn. 56 over the protein volume: It is noted that is 2 . This is the formula for simulation data fitting, referred to as eqn. 3b in the main text by retaining only the first term (n = 1) in the n series which is expected to decay rapidly as n increases.

SI-5 t-r-z dependent solution to thermal diffusion equation in the cylindrical model
We solve the thermal diffusion equation for the cylindrical model with the radius Rc and height Hc by ignoring angle (θ) dependence: Using the solutions to the t-r and t-z dependent equations as illustrated in the preceding sections, we can construct a general solution as The arguments in the exponential term (for τ s in eqn. 60), thermal conductivity and thermal conductance are considered for both the radius and height directions, which are annotated by // and ⊥, respectively. We suppose that the cylinder has radius of Rc and height of Hc. Due to the inversion symmetry of cylinder around the origin, it is enough to consider either of the two boundary conditions for the height direction (we here consider the left one given in eqn. 63).
The right-hand side of eqn. 60 is multiplied by − and is substituted into B.C. for the radius direction: Then, by combining the above eqns. 64 and 65, we obtain Similarly, the right-hand side of eqn. 60 is multiplied by − and is substituted into B.C. for the height direction: Then, by combining eqns. 67 and 68 above, we obtain By substituting eqn. 60 into I.C., we obtain the following equation: Eqn. 70 is operated by ∫ 0 � ′ � 0 with the integration of r from 0 to Rc, then leading by recalling the following formulas: