1st International Electronic Conference on Applied Sciences 10–30 November 2020 Numerical Evaluation of Protein Global Vibrations at Terahertz Frequencies by means of Elastic Lattice Models D. Scaramozzino, G. Lacidogna, G. Piana, A. Carpinteri Department of Structural, Geotechnical and Building Engineering Politecnico di Torino, Italy 1
1st International Electronic Conference on Applied Sciences 10–30 November 2020 Outline of the Work 1. Introduction 2. Elastic Lattice Models (ELMs) for Protein Vibrations 3. Validation of the Numerical Models: B-factors 4. Protein Normal Modes and Biological Mechanism 5. Conclusions and Future Developments 2 1/22
1st International Electronic Conference on Applied Sciences 10–30 November 2020 Protein: Sequence of several different amino acids, with a complex three-dimensional shape and function Primary structure Protein folding Three structural Secondary levels structures Tertiary structure 3 1. Introduction 2/22
1st International Electronic Conference on Applied Sciences 10–30 November 2020 The fundamental paradigm of protein action Sequence Dynamics Function Structure 4 1. Introduction 3/22
1st International Electronic Conference on Applied Sciences 10–30 November 2020 How to study the Structure – Dynamics relationship? Protein Form of Type of Type of analysis representation potentials output Increasing computational Molecular Dynamics Complex semi- Increasing model All atoms Trajectories (MD) empirical complexity efficiency Normal mode analysis Multi-parameter Normal All atoms (NMA) harmonic modes Single- All-atom Elastic Lattice Normal All atoms parameter Model (aaELM) modes harmonic Single- Coarse-grained Elastic Only one node per Normal parameter Lattice Model (cgELM) amino acid modes harmonic 5 1. Introduction 4/22
1st International Electronic Conference on Applied Sciences 10–30 November 2020 Elastic Lattice Model (ELM) From the single bar element… … to the spatial ELM i ( x i , y i , z i ) E i,j , A i,j , L i,j 𝑙 ",$ = 𝐹 ",$ 𝐵 ",$ 𝑀 ",$ j ( x j , y j , z j ) 6 2. Elastic Lattice Models (ELMs) for Protein Vibrations 5/22
1st International Electronic Conference on Applied Sciences 10–30 November 2020 Elastic Lattice Model (ELM) – Finite Element (FE) approach ∗ = 𝐹 ",$ 𝐵 ",$ 1 −1 2x2 stiffness matrix of the elastic bar in the local system 𝐥 𝐣,𝐤 −1 1 𝑀 ",$ 𝑦 $ − 𝑦 " 𝑧 $ − 𝑧 " 𝑨 $ − 𝑨 " 0 0 0 2x6 rotation matrix of the 𝑀 ",$ 𝑀 ",$ 𝑀 ",$ elastic bar, between the 𝐎 𝐣,𝐤 = 𝑦 $ − 𝑦 " 𝑧 $ − 𝑧 " 𝑨 $ − 𝑨 " local and global systems 0 0 0 𝑀 ",$ 𝑀 ",$ 𝑀 ",$ 𝑼 𝐥 𝐣,𝐤 ∗ 𝐎 𝐣,𝐤 𝐥 𝐣,𝐤 = 𝐎 𝐣,𝐤 6x6 stiffness matrix of the elastic bar in the global system 7 2. Elastic Lattice Models (ELMs) for Protein Vibrations 6/22
1st International Electronic Conference on Applied Sciences 10–30 November 2020 Elastic Lattice Model (ELM) – Finite Element (FE) approach 𝛃 𝐣,𝐤 −𝛃 𝐣,𝐤 ∗ 𝐎 𝐣,𝐤 = 𝑼 𝐥 𝐣,𝐤 𝐥 𝐣,𝐤 = 𝐎 𝐣,𝐤 −𝛃 𝐣,𝐤 𝛃 𝐣,𝐤 6 𝑦 $ − 𝑦 " 𝑦 $ − 𝑦 " 𝑧 $ − 𝑧 " 𝑦 $ − 𝑦 " 𝑨 $ − 𝑨 " 6 6 6 𝑀 ",$ 𝑀 ",$ 𝑀 ",$ 6 𝛃 𝐣,𝐤 = 𝐹 ",$ 𝐵 ",$ 𝑦 $ − 𝑦 " 𝑧 $ − 𝑧 " 𝑧 $ − 𝑧 " 𝑧 $ − 𝑧 " 𝑨 $ − 𝑨 " 𝑀 ",$ 6 6 6 𝑀 ",$ 𝑀 ",$ 𝑀 ",$ 6 𝑦 $ − 𝑦 " 𝑨 $ − 𝑨 " 𝑧 $ − 𝑧 " 𝑨 $ − 𝑨 " 𝑨 $ − 𝑨 " 6 6 6 𝑀 ",$ 𝑀 ",$ 𝑀 ",$ 6x6 stiffness matrix of the elastic bar in the global system 8 2. Elastic Lattice Models (ELMs) for Protein Vibrations 7/22
1st International Electronic Conference on Applied Sciences 10–30 November 2020 Elastic Lattice Model (ELM) – Finite Element (FE) approach ∗ 𝐎 𝐣,𝐤 𝐃 𝐣,𝐤 𝑼 𝐎 𝐣,𝐤 𝑼 𝐥 𝐣,𝐤 𝐋 = 8 𝐃 𝐣,𝐤 𝐃 𝐣,𝐤 𝒋,𝒌|𝑴 𝒋,𝒌 =𝒔 𝒅 6x3N expansion matrix of the 3Nx3N stiffness matrix of the ELM elastic bar to reach the dimension of the structural problem 𝐍 𝟐 𝟏 … 𝟏 𝑛 " 0 0 𝟏 𝐍 𝟑 … 𝟏 0 𝑛 " 0 𝐍 𝐣 = 𝐍 = … … … 𝟏 0 0 𝑛 " 𝟏 𝟏 𝟏 𝐍 𝐎 3x3 mass matrix of the i th node 3Nx3N mass matrix of the ELM 9 2. Elastic Lattice Models (ELMs) for Protein Vibrations 8/22
1st International Electronic Conference on Applied Sciences 10–30 November 2020 Elastic Lattice Model (ELM) – Anisotropic Network Model (ANM) 𝜖 6 𝑊 𝜖 6 𝑊 𝜖 6 𝑊 𝐈 𝟐,𝟐 𝐈 𝟐,𝟑 … 𝐈 𝟐,𝐎 ",$ ",$ ",$ 𝜖𝑦 " 𝜖𝑦 $ 𝜖𝑦 " 𝜖𝑧 $ 𝜖𝑦 " 𝜖𝑨 𝐈 𝟑,𝟐 𝐈 𝟑,𝟑 … 𝐈 𝟑,𝐎 $ 𝐈 = 𝜖 6 𝑊 𝜖 6 𝑊 𝜖 6 𝑊 … … … … ",$ ",$ ",$ 𝐈 𝐣,𝐤 = 𝐈 𝐎,𝟐 𝐈 𝐎,𝟑 … 𝐈 𝐎,𝐎 𝜖𝑧 " 𝜖𝑦 $ 𝜖𝑧 " 𝜖𝑧 $ 𝜖𝑧 " 𝜖𝑨 $ 𝜖 6 𝑊 𝜖 6 𝑊 𝜖 6 𝑊 3Nx3N Hessian matrix of the ANM ",$ ",$ ",$ 𝜖𝑨 " 𝜖𝑦 $ 𝜖𝑨 " 𝜖𝑧 $ 𝜖𝑨 " 𝜖𝑨 $ 𝑶 1 ",$ = 𝛿 𝐈 𝐣,𝐣 = − 8 𝐈 𝐣,𝐤 ",$P 6 𝛿 ∝ 𝑊 2 𝑠 ",$ − 𝑠 ",$R 𝑠 𝒌J𝟐,𝒌K𝒋 It can be easily demonstrated that there exists complete consistency between the FE- based ELM stiffness matrix K and the ANM Hessian matrix H 10 2. Elastic Lattice Models (ELMs) for Protein Vibrations 9/22
1st International Electronic Conference on Applied Sciences 10–30 November 2020 Elastic Lattice Model (ELM) – Modal Analysis Non-trivial solution 𝐯 = 𝛆 sin 𝜕𝑢 𝐋 − 𝜕 6 𝐍 𝛆 = 𝟏 |𝐋 − 𝜕 6 𝐍| = 𝟏 𝐋𝐯 + 𝐍 ̈ 𝐯 = 𝟏 Set of 3 N -6 non-zero Set of 3 N -6 non-rigid eigenvectors (mode shapes) 𝛆 𝐨 eigenvalues 𝜕 \ = 2𝜌𝑔 \ 11 2. Elastic Lattice Models (ELMs) for Protein Vibrations 10/22
1st International Electronic Conference on Applied Sciences 10–30 November 2020 Effect of the selected cutoff value on the generated ELM r c = 8Å r c = 10Å r c = 12Å *The C α atoms of the protein are the representative points for each amino acid! r c = 15Å r c = 20Å Lysozyme (PDB: 4YM8) – LUSAS FE software used for the construction of the model 2. Elastic Lattice Models (ELMs) for Protein Vibrations 11/22
1st International Electronic Conference on Applied Sciences 10–30 November 2020 How to set up the values of the axial rigidity EA ? With the B-factors! 65 Lysozyme 60 B-factors are a measure of the Experimental B-factor [Å 2 ] 55 protein flexibility and can be Average value 50 found in the PDB file, as obtained 45 from the X-ray crystallographic 40 experiment 35 30 25 20 40 60 80 100 120 Residue (amino acid) B-factors can also be associated to the normal modes fg 𝜀 ",\ 6 𝐶 " = 8 3 𝜌 6 𝑙 c 𝑈 8 𝜕 \6 \Je 13 3. Validation of the Numerical Models: B-factors 12/22
1st International Electronic Conference on Applied Sciences 10–30 November 2020 How to set up the values of the axial rigidity EA ? With the B-factors! Imposing that the average value of the computed B-factors matches the average value of the experimental ones allows to define the rigidity of the ELM elastic bars Mean length of Stiffness of the mean Model Cutoff (Å) EA (pN) the elastic bar (Å) connection (N/m) A 8 5.71 831 1.455 B 10 7.21 235 0.326 C 12 8.61 124 0.144 D 15 10.59 71 0.067 E 20 13.46 45 0.033 14 3. Validation of the Numerical Models: B-factors 13/22
1st International Electronic Conference on Applied Sciences 10–30 November 2020 How to validate the models? With the B-factors! Model Correlation A 0.57 B 0.67 C 0.66 D 0.69 E 0.72 Correlation coefficients from 57% to 72%! These are very high values if you think how much the model is simplified and how much the physics of the problem is complex! 15 3. Validation of the Numerical Models: B-factors 14/22
1st International Electronic Conference on Applied Sciences 10–30 November 2020 Looking at the 1 st vibration modes… … we find a clear representation of the cleft opening-closing motion, which is known to be the actual biological mechanism of the lysozyme 16 4. Protein Normal Modes and Biological Mechanism 15/22
1st International Electronic Conference on Applied Sciences 10–30 November 2020 Looking at the 2 nd vibration modes… … we find an overall torsional twisting of the lysozyme, still with a significant flexibility in the cleft region 17 4. Protein Normal Modes and Biological Mechanism 16/22
1st International Electronic Conference on Applied Sciences 10–30 November 2020 Does the cutoff parameter affect the mode shapes? 1 st vibration mode Absolute displacements MAC matrix 18 4. Protein Normal Modes and Biological Mechanism 17/22
1st International Electronic Conference on Applied Sciences 10–30 November 2020 Does the cutoff parameter affect the mode shapes? 2 nd vibration mode Absolute displacements MAC matrix 19 4. Protein Normal Modes and Biological Mechanism 18/22
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