Applying the enhanced Craig-Bampton method to equilibrium protein dynamics

Transcription

1 Applying the enhance Craig-Bampton metho to equilibrium protein ynamics *Jaehoon Kim ), Jin-Gyun Kim 2), Giseok Yun 3), an Do-Nyun Kim 4) ),3),4) Department of Mechanical an Aerospace Engineering, Seoul National University, Gwanak-ro, Gwanak-gu, Seoul 5-742, Republic of Korea 2) Mechanical Systems Safety Research Division, Korea Institute of Machinery an Materials, Gajeongbuk-ro 56, Yuseong-gu, Daejeon , Republic of Korea ) max7k@snu.ac.kr ABSRAC Finite element (FE) metho has been successfully applie to analysis of equilibrium protein ynamics because of its high accuracy an computational efficiency compare with the methos base on the full atomistic force fiel. However, even using the FE metho, analysis of macromolecular protein assemblies is still challenging ue to its huge number of DOFs. In orer to hanle this problem, we here apply the enhance Craig-Bampton (CB) metho that was a recently evelope, robust FE moel reuction technique. o illustrate, its performance is investigate by analyzing the molecular structure of the Mile East Respiratory Synrome coronavirus (MERS-CoV) 3C-like protease.. INRODUCION he coarse-graine moeling techniques such as elastic network moel (ENM) (irion, 996) an finite element (FE) moel (Bathe, 28, Kim, 2) have been successfully use for analysis of supramolecular protein assemblies. In particular, FE moeling approach offers an efficient way of incorporating the effect of externally applie mechanical forces or surrouning meia (Bathe, 28, Kim, 2) as it moels the molecular surface explicitly. Nevertheless, analysis of supramolecular protein assemblies is still challenging even with these coarse-graine moeling approaches ue to its huge number of DOFs. o aress this problem, we recently propose an automate moel reuction proceure (Kim an Kim, 25) base on component moe synthesis (CMS) that is a popular reuce orer moeling (ROM) technique in structural ynamics community (Craig an ), 3) Grauate Stuent 2) Senior researcher 4) Professor

2 Bampton, 965, MacNeal, 97, Park, Kim an Lee, 22, Kim, Lee an Lee, 24, Kim an Lee, 24, Kim, Boo an Lee, 25,). In this work, we present an improve proceure that employs the enhance Craig-Bampton ( CB) metho (Kim an Lee, 25). First, we briefly introuce FE base protein moeling an the enhance CB metho. hen, the performance of the enhance CB metho is compare with the original CB metho using numerical examples. 2. MEHOD 2. Finite element moel of proteins Calculating the molecular surfacee of a protein is the first f step inn builing the finite element protein moel. Solvent-exclue surface is generally use as the molecular surface that can be compute by rolling a sphere over the Van er Waals surface. Using PMV version.5.6 ( molecular surface of the protein. Fig. B represents the solvent-exclue surface of an example molecular-viewer) which is freely available, we calculate a triangulate protein obtaine using a sphere raiuss of.5 Å. Fig. (A) Atomic structure, (B) solvent-exclue surface, (C) finite element moel an (D) partitione finite element moel of the Mile East E Respiratory Synrome coronavirus (MERS-CoV) 3C-like protease

3 From the compute molecular surface, we generate the three-imensional volumetric finite element moel using 4-noe tetraheral soli elements as shown in Fig. C, which is one using a commercial finite element analysis program ADINA version 9..7 (ADINA R&D, Inc., Watertown, MA, USA). Finally, the constructe finite element moel is partitione into a number of substructures to be use for application of the enhance CB metho (Fig. D). We use MEIS version 5.. ( for partitioning of the moel. 2.2 Enhance Craig-Bampton metho he original finite element moel of protein is partitione into small substructures using MEIS as in Fig. D, an then the matrices of the equation of motion can be expresse by Μ s Μc K s K c x s Μ, K Μc Μ, x b K c K, () b xb where Μ an K are the mass an stiffness matrices, respectively, an x is isplacement vector. Subscripts s, b an c enote substructural, interface bounary an coupling terms, respectively. In the CB metho, the isplacement vector x can be approximate with the transformation matrix as follows q Φ C x x, x, C K s K c, (2) b Ib in which I b is an ientity matrix of interface bounary, an Φ is a matrix consisting of the substructural ominant normal moes which are the low frequency normal moes obtaine from the substructural eigenvalue problem. It shoul be note that the low frequency normal moes are usually associate with important ynamic characteristics such as the conformational change of proteins. In the CB metho, the reuce matrices are constructe using as Μ p Τ ΜΤ, Κ p Τ ΚΤ. (3) In the enhance CB metho, the enhance transformation matrix is newly erive by consiering the resiual substructural moe as

4 , r (4) with Φ C, Ib r Frs[ M sc Mc] ] M p K p, F rs K s Φ Λ Φ, (5) where F enotes the resiual flexibility, in which containss the resiual moe effect. rs Fig. 2 Comparison between the original an enhance CB methos. (A) eigenvalues, (B) relative eigenvalue errors an (C) RMSFs at alpha-carbons for the Mile East Respiratory Synrome coronavirus (MERS-CoV) 3C-like protease. Finally, the reuce matrices of the enhance CB metho are a obtaine as ~ M p M, K ~ p K. (6)

5 Due to compensation of the resiual substructural moes, the reuce matrices in Eq. (6) are much more accurate than the reuce matrices in Eq. (3). 3. RESUL AND CONCLUSION In this section, we investigate the performance of the enhance CB metho in comparison with the original CB metho. he approximate eigensolutions for the structure of Mile East Respiratory Synrome coronavirus (MERS-CoV) 3C-like protease (RCSB Protein Data Bank, ID: 4WMF) (Neele an Lountos, 25) are calculate by solving the eigenvalue problems using the reuce matrices in Eqs. (3) an (6). Figs. 2A an 2C represent the eigenvalues an the relative eigenvalue errors obtaine using the original an enhance CB methos, respectively. hese results clearly show that the enhance CB metho is more accurate than the original CB metho. In aition, RMSFs (Root-Mean-Square-Fluctuations) at alpha-carbons are presente in Fig. 2C, which illustrates a high correlation between the results calculate using the original an enhance CB methos. ACKNOWLEDGEMENS his research was supporte by the EDucation-research Integration through Simulation On the Net (EDISON) Program (Grant No. 24M3CA638842) an by the Basic Science Research Program (Grant No. 23RAA626) through the National Research Founation of Korea (NRF) fune by the Ministry of Science, IC an Future Planning. REFERENCES Bathe, M. (28) A finite element framework for computation of protein normal moes an mechanical response, Proteins, 7(4), Craig, R.R. an Bampton, M.C.C. (965) Coupling of substructures for ynamic analysis, AIAA J., 6(7), Kim, D.-N., Nguyen, C.-. an Bathe, M. (2) Conformational ynamics of supramolecular protein assemblies, J. Struct. Biol., 73(2), Kim, J.G., Boo, S.H. an Lee, P.S. (25) An enhance AMLS metho an its performance, Comput. Metho Appl. M., 287, 9-. Kim, J., Kim, J.G., Yun, G., Lee, P.S. an Kim, D.-N. (25) owars moular analysis of supramolecular protein assemblies, J. Chem. heory Comput., submitte. Kim, J.G., Lee, K.H. an Lee, P.S. (24) Estimating relative eigenvalue errors in the Craig-Bampton metho, Comput. Struct., 39, 54-64, 24.

6 Kim, J.G. an Lee, P.S. (24) An accurate error estimator for Guyan reuction, Comput. Metho Appl. M., 278, -9. Kim, J.G. an Lee, P.S. (25) An enhance Craig-Bampton metho, Int. J. Numer. Meth. Engng., 3, Kim, J.G., Lee, P.S. an Park, K.C. (25) A moe selection algorithm for the flexibility base component moe synthesis, Proceeings of 5th International Conference on Computational Methos in Structural Dynamics an Earthquake Engineering (COMPDYN 25), Crete, Greece. MacNeal, R.H. (97) Hybri metho of component moe synthesis, Comput. Struct., (4), Neele, D. an Lountos, G.. (25) Structures of the Mile East respiratory synrome coronavirus 3C-like protease reveal insights into substrate specificity, Acta Crystallogr., Sect. D: Biol. Crystallogr., 7(5), 2- Park, K.C., Kim, J.G. an Lee, P.S. (22) A moe selection criterion base on flexibility approach in component moe synthesis, Proceeing 53th AIAA/ASME/ASCE/AHS/ ASC Structures, Structural Dynamics, an Materials Conference, Hawaii, USA. irion, M. M. (996) Large amplitue elastic motions in proteins from a singleparameter, atomic analysis, Phys. Rev. Lett., 77(9), 95.