Principles of pharmacological modelling relevant to applications of imaging endpoints

Talk from the 23 & 24 January 2012 "GlaxoSmithKline - Neurophysics Workshop on Pharmacological MRI", an activity hosted at Warwick University and coordinated with the Neurophysics Marie Curie Initial Training Network of which GSK is a participant

Distribution of the values of torsional angles of a 67 residues long protein (1CTF) during a 3.5 ⋅ 10⁵ time-steps Monte-Carlo simulation.

<p>Each of the stacked barchart is relative to a different torsional angle. At each simulation step the structure is deformed by applying our algorithm to a randomly chosen portion of the chain (with probability 0.7) or by a pivot move (with probability 0.3). In the former case the step size <i>ds</i> is chosen from a normal distribution with mean 0 <i>rad</i> and variance 0.08 <i>rad</i>². In the latter case a randomly chosen <i>ϕ</i> or <i>ψ</i> angle is perturbed by adding to it a quantity that is chosen from a normal distribution with mean 0 <i>rad</i> and variance 0.4 <i>rad</i>².</p

Schematic representation of the portion of a linear chain involved in a local modification.

<p>Bonds (1) and (<i>n</i>_<i>b</i> + 1) are colored in blue while all the others are in red. The degrees of freedom which are varied (<i>ξ</i>₁, …, <i>ξ</i>_<i>n</i>) are arbitrarily distributed inside the region. When they are concertedly changed to new values </p><p></p><p><mo stretchy="false">(</mo></p><p><mi>ξ</mi><mn>1</mn><mo>′</mo></p><mo>,</mo><mo>…</mo><mo>,</mo><p><mi>ξ</mi><mi>n</mi><mo>′</mo></p><mo stretchy="false">)</mo><p></p><p></p>, the new pink configuration is obtained while all the bonds outside the region (in black) remain fixed in space.<p></p

General graphical representation of a chain according to the Denavit-Hartenberg convention, as discussed in the text.

<p>Thick lines represent the physical bonds and spheres the atoms. <i>α</i>, <i>θ</i>, <i>r</i> and <i>d</i> are the DH parameters describing the chain and <i>o</i> is the origin of each local frame . The structure of a portion of a protein backbone chain with all its heavy atoms is shown superimposed.</p

Three-dimensional projection of the solution manifold.

<p>The whole manifold lies in a seven-dimensional space. Some configurations are highlighted: the starting configuration is emphasized with a red circle while two other possible solutions are emphasized in green. The red part of the structure is the portion which has been modified with our moves.</p

Distribution of variability of <i>θ</i>₁ and <i>θ</i>₃ angles for different values of <i>λ</i>₁ for a 9 degrees of freedom chain.

<p>The distribution of <i>θ</i>₃ is largely not affected by the change of <i>λ</i>₁ while the distribution of <i>θ</i>₁ is rescaled according to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0118342#pone.0118342.e072" target="_blank">Eq. (19)</a>.</p

A three-dimensional projection of the solution manifold for our model of cyclooctotetraene molecule (B) and a schematics of how the manifold has been explored (A).

<p>First a set of seven angles are chosen and the relative one-dimensional manifold is visited (red line). Then the one-dimensional manifold relative to a different choice of degrees of freedom is explored using each of the generated structure as a starting point (green dots).</p

Four conformations of the ciclic molecule cyclooctatetraene obtained while exploring its whole conformational space by changing the 8 torsional angles degrees of freedom.

<p>Bond length and bond angles are kept fixed. This is a toy model to illustrate the efficiency of the method: the molecule alternates double and single bonds and therefore half of the torsional angles are constrained and only four are completely free. Images of molecular structures have been generated with PyMol [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0118342#pone.0118342.ref022" target="_blank">22</a>].</p