1. do proteins fold ? Robert Glen Dmitry Nerukh Rimini 2005

2. Proteins…. (Some) Fold into complex 3-dimensional shapes that are reproducible (Some) Fold into complex 3-dimensional shapes that are reproducible Have amino acid sequences that combined with the dynamics of their motion in water implicitly contain information about their observed 3D structure Have amino acid sequences that combined with the dynamics of their motion in water implicitly contain information about their observed 3D structure By a dynamic process in solution, discover these folded forms efficiently (Levinthall paradox) By a dynamic process in solution, discover these folded forms efficiently (Levinthall paradox) What is ‘peculiar’ about the dynamics of such protein sequences (good folders) – how do they ‘know’ how to fold ? What is ‘peculiar’ about the dynamics of such protein sequences (good folders) – how do they ‘know’ how to fold ?

3. Protein dynamics… Can be thought of as a system with ‘emergent’ properties – the structure we observe emerges from the dynamics of the protein in solution Can be thought of as a system with ‘emergent’ properties – the structure we observe emerges from the dynamics of the protein in solution What we want to know: what dynamic processes can we analyse that allow insight into the folding process, particularly the role of solvent? What we want to know: what dynamic processes can we analyse that allow insight into the folding process, particularly the role of solvent?

4. Seemingly random systems have the capacity to generate structured behaviour Seemingly random systems have the capacity to generate structured behaviour –e.g. lots of ants that are similar, communicate locally, yet develop complex communities – and these look surprisingly alike –an ECG pattern from different hearts is similar yet is the result of many interacting cells - a ‘complex’ rhythm develops The interaction between the ‘units’, each other and their environment is creating new information The interaction between the ‘units’, each other and their environment is creating new information This ‘emergent’ behaviour can now be quantified and analysed : for our purposes, we can quantify the emergence of structure in a protein from dynamics This ‘emergent’ behaviour can now be quantified and analysed : for our purposes, we can quantify the emergence of structure in a protein from dynamics Complex systems: the property of emergence

5. Examples of emergence in simple systems The software package StarLogo (from the Media Laboratory, MIT, Cambridge, Massachusetts, free) allows simple systems of self organisation to be constructed. The software package StarLogo (from the Media Laboratory, MIT, Cambridge, Massachusetts, free) allows simple systems of self organisation to be constructed. A couple of examples…. A couple of examples…. 1. Each turtle follows two simple rules: (1) it tries to keep a certain distance from each of its two "neighbors", and (2) it gently "repels" the group as a whole, trying to move away from the other turtles. With these two rules, the turtles arrange themselves into a circle. 1. Each turtle follows two simple rules: (1) it tries to keep a certain distance from each of its two "neighbors", and (2) it gently "repels" the group as a whole, trying to move away from the other turtles. With these two rules, the turtles arrange themselves into a circle. 2. This project explores a simple ecosystem made up of rabbits and grass. The rabbits wander around randomly, and the grass grows randomly. When a rabbit bumps into some grass, it eats the grass and gains energy. If the rabbit gains enough energy, it reproduces. If it doesn't gain enough energy, it dies. The population of rabbits and grass develops an ‘attractor’ – predator/prey balance. 2. This project explores a simple ecosystem made up of rabbits and grass. The rabbits wander around randomly, and the grass grows randomly. When a rabbit bumps into some grass, it eats the grass and gains energy. If the rabbit gains enough energy, it reproduces. If it doesn't gain enough energy, it dies. The population of rabbits and grass develops an ‘attractor’ – predator/prey balance.

6. What do we do that is different ? Contemporary physics can measure order (e.g. temperature) or randomness (e.g. entropy, thermodynamics). There are no tools to address problems of innovation, or the discovery of patterns since there are no physical principles that define and dictate how to measure natural structure. Measuring the computational capabilities of the system is the only way to address such questions: We utilise methods for discovering and quantifying emergence, pattern, information processing and memory capacity in quantitative units. Nerukh D., Karvounis G., Glen R. Complexity of classical dynamics of molecular systems Part 1. methodology. J. Chem. Phys. 117, 9618 (2002) Nerukh D., Karvounis G., Glen R. Complexity of classical dynamics of molecular systems Part 2. Finite statistical complexity of a water-Na + system. J. Chem. Phys. 117, 9611 (2002) Dmitry Nerukh, George Karvounis, and Robert C. Glen, Quantifying the complexity of chaos in multi-basin multidimensional dynamics of molecular systems, Complexity, 10(2), 40-46 (2004)

7. How to quantify complexity We have used a method called computational mechanics (Crutchfield): We have used a method called computational mechanics (Crutchfield): An Information based method using computation theory (‘Shannon entropy and Kolmogorov complexity’) An Information based method using computation theory (‘Shannon entropy and Kolmogorov complexity’) Discovers dynamical patterns Discovers dynamical patterns Emergence is computed from the ability of the system to process information – which means…. Emergence is computed from the ability of the system to process information – which means…. J. P. Crutchfield and K. Young, Phys. Rev. Lett., 63, 105 (1989)

8. What does it really do ? When analysing a trajectory from the past to the future, patterns emerge. To quantify the information in the system, we work out how complex a model is required given the past trajectory, to predict future trajectories – the memory of the system – bigger memory, bigger complexity When analysing a trajectory from the past to the future, patterns emerge. To quantify the information in the system, we work out how complex a model is required given the past trajectory, to predict future trajectories – the memory of the system – bigger memory, bigger complexity This can be quantified by something called an  -machine This can be quantified by something called an  -machine –Below, are two  -machines from transitions observed as a small peptide undergoes a conformational transition – one analysis of the dynamics is obviously more ‘complex’ than the other

9. How do we calculate complexity of real systems ? Firstly, the dynamic trajectory (could be moments of particles, dipole orientation of water etc.) is converted from a time-based signal to a symbolic sequence The we form “equivalence classes”: Past…01010 00000…Future with probability 0.1 01000…Future with probability 0.7 11101…Future with probability 0.2 Calculating complexity: Where K is a number of equivalence classes, P(L i ) is a probability of the i equivalence class (Shannon entropy) We look at the probability of going from one state to another, using the ‘expectation probability’, called the ‘surprise’ in probability theory, and calculate the memory requirements of an ‘e-machine’ that would be required to describe the process. This approach is called computational mechanics (Crutchfield et. al ). The finite statistical complexity is calculated as

10. Why use complexity? You are probably familiar with a free energy funnel – this is misleading – its not 2D! You are probably familiar with a free energy funnel – this is misleading – its not 2D! The complexity approach is designed to quantify the self- organisation in dynamic systems which is multi-dimensional – the funnel is also multi-dimensional and represents not so much an energy funnel, but a description of transitions that utilise quasi- periodic dynamics The complexity approach is designed to quantify the self- organisation in dynamic systems which is multi-dimensional – the funnel is also multi-dimensional and represents not so much an energy funnel, but a description of transitions that utilise quasi- periodic dynamics Theodore L. Brown, Making Truth. The Roles of Metaphor in Science

11. Calculating the complexity of real systems How do we make sense out of this: oxygen atom motions from water close to the protein trajectory

12. Trajectories, dynamics This is a cartoon of the global system This is a cartoon of the global system In reality, it’s a High- dimensional phase space In reality, it’s a High- dimensional phase space Our expectation was that in transitioning from one state to another, complexity would change Our expectation was that in transitioning from one state to another, complexity would change Here we are talking about the change in the dynamic behaviour (could be the protein or solvent for example) of the system Here we are talking about the change in the dynamic behaviour (could be the protein or solvent for example) of the system Like two guitar strings – E moves differently from A. Like two guitar strings – E moves differently from A.

13. What about an example of some simple molecular dynamics showing different dynamic regimes – e.g. two water molecules going from chaotic to quasi-periodic motion

14. 2 waters Initial conditions – chaotic motion Evolve into quasi-periodic motion This is an atractor, a state into which the dynamic system approaches having a single basin of attraction

15. 2 waters 6 degrees of freedom (for one molecule) 6 degrees of freedom (for one molecule) Oxygen Coordinates Orientation of Dipole Change in dynamic regime

16. 2 waters Orientation of Dipole Notice the change in Complexity of two hydrogen atoms of a water molecule as the simulation progresses and goes from chaotic to quasi-periodic motion

17. Trajectories, dynamics Back to a cartoon of the global system Back to a cartoon of the global system Do the transitions happen in a similar ‘concerted’ fashion, with a decrease in complexity of dynamics? Do the transitions happen in a similar ‘concerted’ fashion, with a decrease in complexity of dynamics?

18. Looking at the complexity of a transition from an extended conformation of a small peptide to form a  -turn - Leu Enkephalin

19. (Leu-enkephalin, Karle, I.L., Karle, J., Mastropaolo, D., Camerman, A., & Camerman, N. (1983) Acta Cryst., B39, 625-637.) Enkephalins are Small molecule pentapeptides, found in the brain, have opioid activity. Tyr-Gly-Gly-Phe-(Leu/Met). From nmr data, some structure is seen in solution. We were interested in possible stabilisation of intermittent  -turn motifs. Similations using Gromacs in explicit water (SPC) revealed several turn-like events. Water network dynamics at the critical moment of a peptides b-turn formation: a molecular dynamics study. George Karvounis, Dmitry Nerukh and Robert C. Glen, J. Chem. Phys. 2004, 121, 4925. leu-enkephalin X-ray structure

20.  -turn form How does complexity change for the peptide and for the water? open form

22. enkephalin Formation of a  -turn

23. An example of peptide Dynamic regimes An example of a transition (fast) from one minimum to another for this dihedral angle

24. Complexity analysis of peptide/water molecules during the folding event turn event turn event Topological complexities of the peptide’s atoms. The symbolisation alphabet of size 8 and history length of 3 ps were used. The  -turn transition is at 1657 ps Topological complexities of the waters’ atoms. A symbolisation alphabet of size 32 and history length of 4 ps were used. The  -turn transition is at 1657 ps

25. Topological complexities of the peptide’s atoms. The symbolisation alphabet of size 8 and history length of 3 ps were used. The  -turn transition is at 1657 ps Topological complexities of the waters’ atoms. A symbolisation alphabet of size 32 and history length of 4 ps were used. The  -turn transition is at 1657 ps

26. What’s happening - ? The dynamics of the states between the transitions is significantly chaotic while at the moment of the transition it becomes semi-chaotic or quasi-regular i.e. the system can maintain approximate constants of motion and possess fully deterministic dynamics. We hypothesise that the effect is the manifestation of this phenomenon. The low complexity value in this case corresponds to less chaotic motion. As a simplified illustration of the dynamics, the transition can be visualised as passing through a narrow “tunnel” connecting two states. In this situation the phase- space flow should “straighten” in order to be transferred from one basin to the other. How ‘tight’ is the tunnel ?

27. Sensitivity of folding transitions to small perturbations of the solvent or peptide A larger system. Chignolin, 10 amino acids A larger system. Chignolin, 10 amino acids GLY-TYR-ASP-PRO- GLU-THR-GLY-THR- TRP-GLY Simulated for 1ns and transitions in conformation analysed.

28. chignolin, showing a  -turn nmr-structure 1UAO

29. Chignolin – showing how a small perturbation (lower) to the velocity of one atom prevents a transition (upper) Perturbation applied here – transition doesn’t happen! This can be applied to a water molecule 15Ǻ away from the peptide! So, the ‘tunnel’ is very tight. The whole system is in concerted motion at this time

30. Conclusions ‘Why’ proteins fold can be viewed as a phenomenon of the dynamics of the system ‘Why’ proteins fold can be viewed as a phenomenon of the dynamics of the system Complexity analysis can provide a different perspective Complexity analysis can provide a different perspective Folding transitions follow very tight concerted motions Folding transitions follow very tight concerted motions Small perturbations can disrupt transitions – even up to 15Ǻ away. Small perturbations can disrupt transitions – even up to 15Ǻ away. Caveat: all the dynamics simulations were performed using Gromacs and the results are obviously subject to how accurately the methodology reflects protein dynamics. Caveat: all the dynamics simulations were performed using Gromacs and the results are obviously subject to how accurately the methodology reflects protein dynamics.

31. Acknowledgements George Karvounis, Herman Berendsen, Makoto Taiji George Karvounis, Herman Berendsen, Makoto Taiji Unilever, the Newton Trust, the EPSRC. Unilever, the Newton Trust, the EPSRC. Gromacs: Gromacs: H. J. C. Berendsen, D. van der Spoel and R. van Drunen, GROMACS: A Message- passing Parallel Molecular Dynamics Implementation, Comp. Phys. Commun., 91, 43-56 (1995) GROMOS W. R. P. Scott, P. H. Hunenberger, I. G. Tironi, A. E. Mark, S. R. Billeter, J. Fennen, A. E. Torda, T. Huber, P. Kruger, W. F. van Gunsteren, The GROMOS Biomolecular Simulation Program Package, J. Phys. Chem. A, 103, 3596-3607 (1999) H. J. C. Berendsen, D. van der Spoel and R. van Drunen, GROMACS: A Message- passing Parallel Molecular Dynamics Implementation, Comp. Phys. Commun., 91, 43-56 (1995) GROMOS W. R. P. Scott, P. H. Hunenberger, I. G. Tironi, A. E. Mark, S. R. Billeter, J. Fennen, A. E. Torda, T. Huber, P. Kruger, W. F. van Gunsteren, The GROMOS Biomolecular Simulation Program Package, J. Phys. Chem. A, 103, 3596-3607 (1999)

33. ECG data in collaboration with Addenbrookes hospital Dr IB Wilkinson Clinical Pharmacology Unit, Addenbrooke's Hospital, Cambridge Complexity for ‘normal’ heart rhythm Change in complexity upon drug dosing Moving the subject drugged heart (continuation of red curve)