Non-Gaussian statistics of the vibrational fluctuations of myoglobin

Abstract

Experiments on the dynamics of vibrational fluctuations in myoglobin revealed an interesting behavioral cross-over occurring in the range 180–200 K. In this temperature range the mean square displacement of atomic positions versus temperature sharply increases its slope, indicating the dissociation of CO from the heme group. In this paper we develop a theoretical model that provides a framework for the quantitative description of this phenomenon. The basis of our calculations is an assumption of an effective potential with multiple local minima. In particular, we consider a quartic potential in place of the simple quadratic. We then use non-Gaussian statistics to obtain a relationship between the mean square displacement and model parameters. We compare our model to published experimental data and show that it can describe the data set using physically meaningful parameters which are fitted to the experimental data. In the process we verify the Gaussian approximation's applicability only to the low-temperature régime. In the high-temperature limit, however, deviations from the Gaussian approximation are due to the double-well nature of our effective potential. We find that the published datasets showing the thermal transition display the qualitative trends predicted by appropriate algebraic approximations to our predicted myoglobin behavior.

Appendix

Appendix A: derivation of potential

In this appendix we show how to fix the three parameters a, b, and c to provide a unique potential. We consider Eqs. (1), (6), V(x _A)=d _A (by definition), and choose x _A<x _B; it is then possible to obtain by algebraic manipulation:

$$ a = \frac{{c^2 - w^2 }} {{12d_{\text{A}} }} $$

(A1)

$$ b = \sqrt {\frac{{2(c - w)(2c + w)^2 }} {{27d_{\text{A}} }}} $$

(A2)

Further algebra and the change of variable c=fw produces:

$$ r = \frac{{d_{\text{B}} }} {{d_{\text{A}} }} = \frac{{f^3 (f + 2)}} {{(f - 1)(f + 1)^3 }} $$

(A3)

We are interested in the case that 0≥r≥1, r $\in$ R. It can be shown that while in general there are four roots for f, only one:

$$ f = - \left( {\frac{1} {2}} \right) + \frac{1} {2}\sqrt {1 + \frac{{2^{2/3} r^{1/3} }} {{(r - 1)^{2/3} }}} - \frac{1} {2}\sqrt {2 - \frac{{2^{2/3} r^{1/3} }} {{(r - 1)^{2/3} }} + \frac{{ - 8 + \frac{{16r}} {{r - 1}}}} {{4\sqrt {1 + \frac{{2^{2/3} r^{1/3} }} {{(r - 1)^{2/3} }}} }}} $$

(A4)

corresponds to the case we are interested in and lies in the range [0, −1/2].

Appendix B: derivation of <x ²>

Each of the integrals in Eq. (10) can be evaluated using the formula (Tuszyński et al. 1985, 1986; Tuszyński and Wierzbicki 1991):

$$ \int\limits_0^\infty {y^{2mp - 1} {\text{e}}^{ - By^{2m} - Fy^{4m} } {\text{d}}y} = \left( {2m} \right)^{ - 1} \left( {2F} \right)^{ - p/2} \Gamma \left( p \right)D_{ - p} \left( {B\left( {2F} \right)^{ - 1/2} } \right){\text{e}}^{B^2 /2F} $$

(B1)

where D _−p is a parabolic cylinder function and Γ(p) a standard Gamma function (Abramowitz and Stegun 1972). Applying this method we obtain:

$$ I_2 = {\text{e}}^{ - \beta V_0 } {\text{e}}^{\beta ^2 c^2 /2\beta a} \sum\limits_{q = 0}^\infty {\frac{1} {{\left( {2\beta a} \right)^{\frac{{3 + 6q}} {4}} }}\Gamma \left( {\frac{{3 + 6q}} {2}} \right)} D_{ - \frac{{3 + 6q}} {2}} \frac{{\beta c}} {{\sqrt {2\beta a} }}\frac{{\left( {\beta b} \right)^{2q} }} {{\left( {2q} \right)!}} $$

(B2)

and:

$$ I_0 = {\text{e}}^{ - \beta V_0 } {\text{e}}^{\beta ^2 c^2 /2\beta a} \sum\limits_{q = 0}^\infty {\frac{1} {{\left( {2\beta a} \right)^{\frac{{1 + 6q}} {4}} }}\Gamma \left( {\frac{{1 + 6q}} {2}} \right)} D_{ - \frac{{1 + 6q}} {2}} \frac{{\beta c}} {{\sqrt {2\beta a} }}\frac{{\left( {\beta b} \right)^{2q} }} {{\left( {2q} \right)!}} $$

(B3)

The parabolic cylinder function has two well-known asymptotic limits, namely for $x \approx 0$:

$$ D_{{ - a - \tfrac{1} {2}}} {\left( { \pm x} \right)} \approx {\text{e}}^{{ \mp {\sqrt {ax} }}} {\left[ {\frac{{{\sqrt \pi }2^{{ - \frac{a} {2} - \frac{1} {4}}} }} {{\Gamma {\left( {\frac{3} {4} + \frac{a} {2}} \right)}}}} \right]} $$

(B4)

and for x>>1:

$$ D_{ - a - \tfrac{1} {2}} \left( x \right) \approx x^{ - a - \tfrac{1} {2}} {\text{e}}^{ - x^2 /4} $$

(B5)