Elastic models of conformational transitions in macromolecules Moon K. Kim, Gregory S. Chirikjian, Robert L. Jernigan Presented by: Amit Jain NUS CS5247

Outline Introduction Problem Statement Related Work Elastic Model Approach Results Conclusion NUS CS5247

Introduction What are Macromolecules ? Carbohydrates Proteins Nucleic Acids Why are conformational transitions important ? Understanding functional relationship Macromolecules are molecules with large molecular sizes as compared to normal molecules. Normally usage is restricted to polymers. Here we are interested in biological macromolecules like proteins Functional relationships like catalysts, etc. in which the transitionary conformation is useful NUS CS5247

Problem Statement Input: Two conformation x, χ Output: List of realistic displacements δ NUS CS5247

Related Work Interpolation Molecular Dynamics Coarse models Linear interpolation in cartesian space Linear interpolation in bond lengths and bond angles Molecular Dynamics Coarse models Simple Unrealistic bond lengths and bond angles Solves the unrealistic bond lengths and angles Can still get impossible paths Linear interpolation by Vonrhein Impossible paths because: the bond lengths and angles are supposed to be fixed in interpolation but they can be different in the conformations themselves. 2: since interpolation is local it might create high energy molecules Molecular Dynamics: propogate the dynamics and wait for it to reach the destination conformation. Computationally very expensive Coarse Models: model only the backbone and a simplified energy function in order to aproximate the whole thing. Good for global motion prediction. NUS CS5247

Outline Introduction Problem Statement Related Work Elastic Model Approach Results Conclusion NUS CS5247

Elastic Model Key Enabler Model Discrete model for small conformational changes about an equilibrium position for macromolecules Model Alpha carbons Inter carbon force interactions Alpha carbons: x,y,z Origin at COM, axis along moment of inertia. Force interactions are springs = kx, where k is 1 when distance smaller than some threshold. Explain the concept of springs having forces stored in change from equilibrium positions Sparcity and reality maintained in motion by keeping a constant number of neighbors starting from closest to farthest. NUS CS5247

Elastic Model Potential Energy function Can be reduced to NUS CS5247

Elastic Model Substituting, We get, Energy is crucial for an elastic model, stable configurations have low energy Spring energy functions are of this form Significance of this change is that this provides a cheap way of calculating energy about a stable configuration, useful for carrying out coarse grained NMA. This is an important analysis tool in calculating analytical motions of macromolecules. Values of k are such that the same expression comes about … NUS CS5247

Elastic Model Analytical equation for motion where M is diagonal matrix If masses are equal then eigenvectors of K are the normal modes, Normal mode analysis is a common way to evaluate potential protein motions NUS CS5247

Approach Interpolate conformation Find the displacement(δ) which minimizes cost Increment alpha Calculate new Use new conformation to find new displacement Cost means:my potential energy if target (l_i,j) is a stable state NUS CS5247

Approach Cost can be simplified to, Hence, minimizing for δ Info: Solution to the displacement is not unique, And so a linear momentum constraint is added Masses are assumed to be same. So only, \sum_i \delta_i = 0 NUS CS5247

Approach Issue: Solution: Position and orientation of final conformation does not match Solution: Superposition RMS Superposition Incremental Rotation Does not match because: we are going around dealing only with the inter molecule distance as constraints. We want them to match because we would like to verify that the process has happened correctly in an automated way. RMS superposition: compares the two conformations and gives a one shot rotation answer Incremental Rotation: calculates the rotation for each intermediate conformation. By minimizing the rotational error. Incremental is more efficient but need to generate all intermediate conformations NUS CS5247

Outline Introduction Problem Statement Related Work Elastic Model Approach Results Conclusion NUS CS5247

Results NUS CS5247

Results NUS CS5247

Results: Comparison with NMA Consistent with NMA analysis NUS CS5247

Conclusion Elastic network model has been introduced Benefits Efficient and realistic simulation of macromolecules transitions Faster NMA analysis (Coarse grained) NUS CS5247

Questions NUS CS5247

Thank You NUS CS5247