Applying Computational Dynamics method to Protein Elastic Network Model

Transcription

1 Applying Computational Dynamic method to Protein Elatic Network Model *Jae In Kim ), Kilho Eom 2) and Sungoo Na 3) ), 3) Department of Mechanical Engineering, Korea Univerity, Seongbuk-gu, Anamdong, Seoul 36-73, Korea 2) College of Sport Science Sungkyunkwan Univerity, Suwon , Republic of Korea ) jay44@korea.ac.kr 2) kilhoeom@kku.edu 3) na@korea.ac.kr ABSRAC Normal mode analyi (NMA) i an effective method for analyzing the dynamic of protein tructure. Elatic network model (ENM) i a harmonic potential baed imple model to decribe protein dynamic. ENM baed NMA i a widely adopted method due to it implicity and robutne for variou protein. Depite the implicity of ENM, till long computation time i needed for upra molecular protein tructure. We extended ENM with computational dynamic method uch a condenation (Kim, Jang et al. 2009) and component mode ynthei (Kim, Eom et al. 2008, Kim, Na et al. 2009). hee method allowed to reduce computation time keeping the accuracy of predicting tructural meaure a imilar a ENM.. INRODUCION Protein dynamic i eential for protein related to it function. Protein in living bodie perform it own function, and thee function are related to binding ligand, tranporting a cargo, propagating ignal, etc.(bahar and Rader 2005, Bahar, Lezon et al. 200) Protein exhibit it own motion related to their function. herefore undertanding protein motion i an important concern a undertanding protein function. Experimental method uch a X-ray crytallography and NMR, cryo-em enable to find molecular tructure of protein, however they find only ingle tate of protein model. o undertand the pathway of conformational change, other method mut be conidered. But it i hard to find the molecular detail of conformational change of protein. Simulation method are effective for undertanding the conformational change of ) Reearch Profeor 2) Profeor 3) Profeor

2 protein in atomic level. Conventionally molecular dynamic (MD) imulation i applied for undertanding protein dynamic.(izrailev, Stepaniant et al. 998, Doruker, Atilgan et al. 2000) However, MD conider complicated potential energie and all-atom information of protein. Moreover, water-box mut be added to imulate, which increae the total number of atom. o overcome thi computational problem of large reource, normal mode analyi (NMA) i applied.(go, Noguti et al. 983, Cui and Bahar 2006) At firt, NMA wa performed with all-atom tructure. However, irion uggeted that a imple harmonic potential energy may decribe the atomic interaction of protein well.(irion 996) Bahar et.al uggeted a model which conider only α-carbon from protein tructure with harmonic potential energy, which i elatic network model (ENM).(Haliloglu, Bahar et al. 997, Atilgan, Durell et al. 200) ENM i an efficient method to undertand protein dynamic. However in cae conidering large protein, ENM could alo require large computational reource. o overcome thi problem variou reduction method were introduced.(ming, Kong et al. 2003, Ming, Kong et al. 2003, Kim, Jernigan et al. 2005) In tructural dynamic view, component mode ynthei (CMS) i a very efficient method maintaining the accuracy of reult and reducing computational time. We applied CMS to ENM baed protein tructure. CMS applied on protein tructure decribed accurate eigenvalue compared to conventional ENM. 2. Elatic network model A mentioned in introduction, ENM i a imple model to decribe protein. ENM decribe amino-acid of protein with a ingle peudo atom on the poition of α-carbon atom.(haliloglu, Bahar et al. 997, Atilgan, Durell et al. 200) hi i an eential atom for amino-acid which play a role a backbone of protein. Atomic coare-graining lead to coare-graining of potential energy.(ozzini 2005) In contrat to all-atom model, ENM conider imple harmonic potential for interaction between α-carbon atom. he potential energy i written a following V () here, γ i force contant of harmonic interaction, N i number of atom, r ij i the ditance of two atom i and j, and upercript 0 tand for the equilibrium conformational tate. Force contant i contant for all interaction of atom. Cut-off ditance r c i introduced to determine the interaction range of atom. Uually r c =7~2Å i applied for ENM to decribe the fluctuation of protein well. Stiffne matrix Γ i evaluated uing the cut-off ditance condition. Γ, Γ, 0,, (2) NMA of ENM provide the conformational fluctuation of protein, whoe motion i decribed by eigenvalue problem

3 where ω i the natural frequency, and q i i it normal mode. 3. Component Mode Synthei γγ (3) o begin with uing CMS, we conider the tructure a an aemblage of ubtructure.(meirovitch 980, Craig 98) CMS decribe the motion eparately over each of the ubtructure by generating a eigenvalue problem and then aemble the ubtructure to work together a a ingle combined tructure by uing geometric compatibility at the interface of adjacent ubtructure. Since each ubtructure i modeled eparately, there are redundant coordinate, a atom hared by two adjacent ubtructure behave the ame motion. he removal of redundant coordinate i carried out during an aembling proce in which the contituent ubtructure are contrained to act a a whole tructure. In the proce, we calculate the mode of each component uing NMA. Subtructure are aembled uing contraint ma point, and the mode of aembled tructure can be compoed. Uing thi methodology, we can approach the protein dynamic in a view point of domain, while maintaining computational accuracy, in term of thermal fluctuation, and eigenolution. Firt, we conider a Hamiltonian for a given protein tructure. H M K (4) N N u u u u 2 2 u i a phyical diplacement vector for an arbitrary point P on the ubtructure, M and K are ma and tiffne matrice for ubtructure. Phyical diplacement vector u i repreented a erie of pace-dependent function multiplied by time-dependent generalized coordinate a below: u ( Pt, ) Ψ ( P) ( t) (5) where Ψ may be regarded a aumed mode vector of ubtructure, and Hamiltonian can be rewritten a W 0 0 E 0 0 H N N W N 0 0 N E N N W E 2 2 (6) Where, W and E are matrice aociated with eigenvector W Ψ M Ψ, repectively. (7)

4 and Relating with, E W and E are block-diagonal matrice of E Ψ K Ψ become diagonal matrix of eigenvalue. W and A vector i defined a uch that the degree of freedom of a vector i N given by n. Vector include a certain number of redundant value due to contraint of boundarie between adjacent component. Auming there are n c contraint and ha n dimenion, total dimenion i n n n. c Coordinate tranformation between () t and () t i related a () t B () t (9) where B i a contraint matrix, and () t i the n-dimenional independent generalized coordinate vector. he matrix B i n n and reflect certain geometric E. (8) condition at boundary point for which are hared by ubtructure r and, becaue the point ha the ame diplacement a below u u r r (0) Moreover, the tranlational diplacement at the interface are related to the generalized diplacement vector () t. In view of thi, one can combine Eq.(0) correponding to all interface into a ingle contraint equation a 0 () hen dividing the vector () t into vector () t of independent variable and a vector p of dependent variable and partitioning the matrix a follow: = 2 (2) Eq.() can be rewritten a p 2 0 (3) which yield p2 (4) Eq.(4) permit one to write a relation Eq.(9) between the () t of independent generalized coordinate for the full tructure and the vector. I B (5) 2 Where I i a unit matrix of order n. he contraint condition allow one to integrate the component to aembled tructure. Now we can rewrite the Hamiltonian for aembled protein tructure. H Q O (6) 2 2

5 where Q B WB, O B are n matrice for the aembledd tructure. n Uing thee Q and O matrice lead to eigenvalue problem OU QU (8) U i n n modal matrix and i diagonal matrix of eigenvaluee e of recontructed tructure. Here we ue the relation Φ BU (9) Here i defined in imilar way with, which obtain n matrix, whoe column vector give the atomic diplacement of each mode of the recontructed tructure. hi atomic diplacement can be compared withh eigenvector of NMA of the original tructure. 4. Reult 4. Low frequency normall mode Firt we conider the lowet frequency mode of the protein tructure, which i known a it play a role in protein dynamic. In figure, we w how thee lowet frequency mode for both original ENM tructure and tructure from CMS. C Reult i plotted for F0 APae motor protein (PDB code: c7) E B n (7) Figure. he lowet frequency mode of target protein from thee component mode ynthei tructure and original ENM tructure: F0 APae motor protein (PDB code: c7) 4.2 Mode Correlation Mode Correlation how directt compare of normal mode of two method quantitatively. Here, we compared normal mode from original ENM tructure and CMS tructure in figure 2. Calculated value vary from 0 to, and maller value how bright color and large value how dark color. For the reult, 50 lowet frequency mode are hown for each target protein, and they are howingg very imilar mode for the

6 lowet 0 mode. However, we have hown ome factor like meann quare fluctuation and cro correlation which ue all mode of each tructure howing very good reult. Depite of unequalne off higher mode, other fact howed good reult. hi indicate that the low frequency mode play an importantt role for dynamic of protein tructure. Figure 2. Mode Correlation of target protein from thee component mode ynthei tructure and original ENM tructure: F0 APae motor protein (PDB code: c7) 5. CONCLUSIONSS In thi tudy, we approached the thermal fluctuation for protein with CMS, which conider the tructure a an aembly of a number of ubtructure. We divided the original protein tructure to ubtructure refer to domain of original protein tructure, and applied CMS to aemble the tructure. During D the proce, tiffne matrix wa diagonalized, and high frequency mode weree reduced by the number of contraint. Alo, CMS wa poible to predict with not only 2 ubtructure but even with 3 ubtructure for F0 APae motor protein. Selected S protein were not retricted to repeated tructure a actinn filament(ming, Kong et al. 2003, Ming,, Kong et al. 2003), but CMS wa able to aemble the ubtructure in different configuration. he reult were compared with the imulation data from original ENM. Original ENM tructure wa howing a very good reult for imulating thermal t fluctuation of protein tructure. And the imulation reult from applying CMS to t protein tructure howed coincidence with the original ENM tructure. CMS ha ome benefit due to reducing high frequency normal mode and implifying tiffne matrix into diagonal term, and thi may how a way for imulating large protein tructure which cannot be done by original ENM.

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