Transforming between discrete and continuous angle distribution models: application to protein χ₁ torsions

Abstract

Two commonly employed angular-mobility models for describing amino-acid side-chain χ₁ torsion conformation, the staggered-rotamer jump and the normal probability density, are discussed and performance differences in applications to scalar-coupling data interpretation highlighted. Both models differ in their distinct statistical concepts, representing discrete and continuous angle distributions, respectively. Circular statistics, introduced for describing torsion-angle distributions by using a universal circular order parameter central to all models, suggest another distribution of the continuous class, here referred to as the elliptic model. Characteristic of the elliptic model is that order parameter and circular variance form complementary moduli. Transformations between the parameter sets that describe the probability density functions underlying the different models are provided. Numerical aspects of parameter optimization are considered. The issues are typified by using a set of χ₁ related ³ J coupling constants available for FK506-binding protein. The discrete staggered-rotamer model is found generally to produce lower order parameters, implying elevated rotatory variability in the amino-acid side chains, whereas continuous models tend to give higher order parameters that suggest comparatively less variation in angle conformations. The differences perceived regarding angular mobility are attributed to conceptually different features inherent to the models.

Appendix: Fitting range-bound probability parameters

The following devises a method for optimizing sets of interdependent probability parameters bounded on the interval [0, 1], so as to remain meaningful in their application to averaging distinct states.

Let the observed J coupling constant represent an average due to a weighted superposition of three point distributions in circular torsion-angle space, with fixed directions and associated coupling values given as

$$ \langle J\rangle = p_{1} J_{1} + p_{2} J_{2} + p_{3} J_{3}. $$

(35)

The three dihedral-angle states, may—but do not have to—coincide with the staggered-rotamer conformations, in which case the identities $p_{1} = P_{{g^{+}}}$, p ₂ = P _t , $p_{3} = P_{{g^{-}}}$, and likewise for J, apply as follows,

$$ \langle J\rangle = P_{{g^{+}}} J_{{g^{+}}} + P_{t} J_{t} + P_{{g^{-}}} J_{{g^{-}}}. $$

(36)

The normalising condition, Σ _k p _k  = 1, constrains the value of the terminal probability such that only two independent probability parameters need be determined.

However, two similarly designed parameters p ₁ and p ₂ would be mathematically and numerically interchangeable and, importantly, would not adhere to the normalisation condition, resulting at times in spurious negative probabilities for p ₃. It may therefore be desirable to contain the effective value range of each probability parameter p in an interval [0, a], which is accomplished by parameter transformation employing the logistic sigmoid,

$$ p = a(1 + e^{{- p^{\prime}}})^ {-1}. $$

(37)

Here, an external programme control variable p′ can conveniently be maintained on the unbounded interval [−∞, +∞]. Change in the constrained variable p as a result of change in the unconstrained variable p′ (notice the prime) is then expressed by the derivative

$$ dp/dp^{\prime}=p(a-p) $$

(38)

where the complementary probability $(a-p) = a (1 + e^{{-p^{\prime}}})^{- 1} e^{{-p^{\prime}}}$. Applying the chain rule, the unconstrained probability parameter affects the model function (Eq. 35) according to dJ/dp′ = dJ/dp × dp/dp′.

Let a = 1. Ideally, the primary parameter p ₁ would explore the whole interval [0, 1], while the secondary parameter p ₂ would fall in the reduced interval [0, 1 − p ₁]. Letting p ₂ = p ₂(p ₁) and p ₃ = 1 − p ₁ − p ₂, the model of Eqs. 35 and 36 is re-defined to feature the sought auto-normalising property as follows,

$$ \begin{aligned} \langle J\rangle &= p_{1} J_{{g^{+}}} + p_{2} (1-p_{1})J_{t} + \left\{{1-p_{1} -p_{2} (1-p_{1})} \right\}J_{{g^{-}}} \\ &= p_{1} \left({J_{{g^{+}}} -J_{{g^{-}}}} \right) + p_{2} (1-p_{1})\left({J_{t} -J_{{g^{-}}}} \right) + J_{{g^{-}}} \\ \end{aligned} $$

(39)

The actual probability variables, signified in Eq. 36 by uppercase P _i , that need be established at some stage in the procedure are thus obtained from the constrained probability parameters as

$$ \begin{aligned} P_{{g^{+}}} & = p_{1} \\ P_{t} & = (1-p_{1})p_{2} \\ P_{{g^{-}}} & = (1-p_{1})(1-p_{2}) \\ \end{aligned} $$

(40)

The appearance of product probabilities in Eq. 40 commands the re-definition of all derivatives with respect to each probability, such that the change in function value J with respect to two unconstrained probability parameters is expressed entirely in terms of two constrained probability variables according to

$$ \begin{aligned} dJ/dp_{1}^{\prime} &= p_{1}(1-p_{1})\left({J_{{g^{+}}} -J_{{g^{-}}}} \right)-p_{1} p_{2}(1-p_{1})\left({J_{t} -J_{{g^{-}}}} \right) \\dJ/dp_{2}^{\prime}&= p_{2} (1-p_{1})(1-p_{2})\left({J_{t}-J_{{g^{-}}}} \right). \end{aligned}$$

(41)

Expressed in terms of the three actual probabilities the derivatives read

$$ \begin{aligned}dJ/dp_{1}^{\prime} &= P_{{g^{+}}}\left({1-P_{{g^{+}}}} \right)\left({J_{{g^{+}}} -J_{{g^{-}}}}\right)-P_{{g^{+}}} P_{t} \left({J_{t} -J_{{g^{-}}}} \right) \\ dJ/dp_{2}^{\prime} &= P_{t}P_{{g^{-}}} \left({{1-P_{{g^{+}}}}} \right) ^{- 1}\left({J_{t}-J_{{g^{-}}}} \right) \end{aligned} $$

(42)

where identities of Eq. 40 were used and, resulting from these, $(1-p_{2}) = P_{{g^{-}}}/(1-P_{{g^{+}}})$. Written as

$$ \begin{aligned} P_{{g^{+}}} & = \left({1 + e^{{{-p^{\prime}_{1}}}}} \right)^{- 1} \\ P_{t} & = \left({1-P_{{g^{+}}}}\right)\left({1 + e^{{{-p^{\prime} _{2}}}}} \right)^{- 1} \\P_{{g^{-}}} & = 1-P_{{g^{+}}} -P_{t} \\ \end{aligned} $$

(43)

the re-defined parameter transformations, exempt of all auxiliary variables p _i , exclusively require the unconstrained parameters $p_1^{\prime}$ and $p_2^{\prime}$ be parsed to the optimizer engine.

It is noteworthy that the first derivative in Eqs. 41 and 42 involves two coupling differences between two pairs of staggered-rotamer states, whereas the second derivative involves only one such difference. This is consistent with the notion that one fewer degree of freedom is left once the first probability has been assigned. Assigning the second probability obviates the need for any further coupling information as the remaining third probability is fully determined.

The difference couplings are given solely by the phase offset associated with the respective coupling type

$$ \left({J_{{g^{+}}} -J_{{g^{-}}}} \right) = -(3)^{1/2} (C_{1} \sin \Updelta \theta + C_{2} \sin 2\Updelta \theta) $$

(44a)

$$ \left({J_{t} -J_{{g^{-}}}} \right) = -(3/4)^{1/2} (C_{1} \sin \Updelta \theta + C_{2} \sin 2\Updelta \theta)-(3/2)(C_{1} \cos \Updelta \theta -C_{2} \cos 2\Updelta \theta) $$

(44b)

where C ₁ and C ₂ are Karplus coefficients appropriate for the coupled pair of nuclei in the χ₁ torsion-angle topology considered, and Δθ is the phase increment to χ₁ associated with that particular coupled spin pair.

For symmetric Karplus curves, the gauche couplings $J_{{g^{+}}}$ and $J_{{g^{-}}}$ are identical, thereby effectively removing the first term from the derivative $dJ/dp_{1}^{\prime}$. If asymmetric curves are to be considered that include a first-order sine component (Schmidt 2007b), the above Eqs. 44a and 44b are extended by the terms −(3)^1/2 S ₁ cos Δθ and −(3/4)^1/2 S ₁ cos Δθ + (3/2) S ₁ sin Δθ, respectively.