Graphical models for protein structure: Difference between revisions

Revision as of 03:56, 12 February 2019

Revision as of 15:35, 11 March 2019

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Where <math>Z = (2\pi)^{n/2}|\Sigma|^{1/2}</math> is the closed form for the [[Partition function (mathematics)|partition function]]. The parameters of this distribution are <math>\mu</math> and <math>\Sigma</math>. <math>\mu</math> is the vector of [[mean values]] of each variable, and <math>\Sigma^{-1}</math>, the inverse of the [[covariance matrix]], also known as the [[precision matrix]]. Precision matrix contains the pairwise dependencies between the variables. A zero value in <math>\Sigma^{-1}</math> means that conditioned on the values of the other variables, the two corresponding variable are independent of each other.

To learn the graph structure as a multivariate Gaussian graphical model, we can use either [[L-1 regularization]], or [[neighborhood selection]] algorithms. These algorithms simultaneously learn a graph structure and the edge strength of the connected nodes. An edge strength corresponds to the potential function defined on the corresponding two-node [[Clique_Clique (graph_theorygraph theory)|clique]]. We use a training set of a number of PDB structures to learn the <math>\mu</math> and <math>\Sigma^{-1}</math>.

Once the model is learned, we can repeat the same step as in the discrete case, to get the density functions at each node, and use analytical form to calculate the free energy. Here, the [[Partition function (mathematics)|partition function]] already has a [[Closed-form expression|closed form]], so the [[inference]], at least for the Gaussian graphical models is trivial. If the analytical form of the partition function is not available, [[particle filtering]] or [[expectation propagation]] can be used to approximate ''Z'', and then perform the inference and calculate free energy.

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* http://www.learningtheory.org/colt2008/81-Zhou.pdf

* {{cite journal|author1= Liu Y |author2= Carbonell J |author3= Gopalakrishnan V |year=2009|title= Conditional graphical models for protein structural motif recognition

|journal= J. Comput. Biol. | volume=16|pages= 639–57 |doi=10.1089/cmb.2008.0176 |pmid=19432536 |issue=5}}

* [http://www.cs.cmu.edu/~jgc/publication/Predicting_Protein_Folds_ICML_2005.pdf Predicting Protein Folds with Structural Repeats Using a Chain Graph Model]