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Protein Folding 2 Thermodynamics Lattice Model Design of foldable proteins Calculations of Z-score Lattice Model • Amino acids sit in the nodes of the cubic lattice. • They interact with their neighbors on the lattice. • It’s very hard to fold such protein: LEVINTHAL PARADOX 1 Energy function Energy function: Potential: U(x,y) Sequence: ai L D Structure: A P S Q I E V K i j if i and j are in contact otherwise Monte Carlo Dynamics 1. Pick a fragment. 2. Try a move 3. Compute ΔE 4. Accept the move with probability P P=1 if "E ! 0 P=exp(-ΔE/T) if ΔE>0 •What do you expect to see at low T? high T? 2 Demo Evolving a foldable sequence = sequence with a large energy gap Three approaches to obtain foldable sequences: En ! E Ec ! E 1. Maximize 2. Minimize 3. Minimize En, while constraining amino acid composition (i.e. σ and <E>) Z= En ! E " (E) ! ! E = E(a, !,U ); En = E(a, ! n ,U ); Algorithm: Metropolis Monte Carlo (for case 3: minimize the energy in the native structure) 1. Pick the Native Structure, a starting (random) sequence, and a potential U(x,y) 2. Mutate the sequence and calculate the change in En : ΔEn=En(a’)-En(a) 3. Accept or reject the mutation: accept with Prob=max{1,exp(-ΔEn/Tseq)} 4. Go to step 2. This satisfies Detailed Balance: ! ! ! ' $ E(a!" ) # E(a) P(a ! a" ) ) ! ! = exp & # P(a" ! a) Tseq &% )( Conversing to equilibrium ! % " E(a) ! P(a) = exp $ ! ' $# Tseq '& LDA P S A I E V K LDA P S Q I E V K 3 Evolved proteins can fold! Random conformation Folding Time: <t>=106-108 for 48-mer <t>=105-107 for 27-mer Conformation of minimal energy Folding time depends on the sequence and structure of the protein! 4