do proteins fold ?do proteins fold ?

Robert GlenDmitry Nerukh

Rimini 2005

Proteins….Proteins….• (Some) Fold into complex 3-dimensional (Some) Fold into complex 3-dimensional

shapes that are reproducibleshapes that are reproducible

• Have amino acid sequences that Have amino acid sequences that combined with the dynamics of their combined with the dynamics of their motion in watermotion in water implicitly contain implicitly contain information about their observed 3D information about their observed 3D structurestructure

• By a dynamic process in solution, discover By a dynamic process in solution, discover these folded forms efficiently (Levinthall these folded forms efficiently (Levinthall paradox)paradox)

• What is ‘peculiar’ about the dynamics of What is ‘peculiar’ about the dynamics of such protein sequences (good folders) – such protein sequences (good folders) – how do they ‘know’ how to fold ?how do they ‘know’ how to fold ?

Protein dynamics…Protein dynamics…

• Can be thought of as a system with Can be thought of as a system with ‘emergent’ properties – the structure ‘emergent’ properties – the structure we observe emerges from the we observe emerges from the dynamics of the protein in solutiondynamics of the protein in solution

• What we want to know: what dynamic What we want to know: what dynamic processes can we analyse that allow processes can we analyse that allow insight into the folding process, insight into the folding process, particularly the role of solvent?particularly the role of solvent?

• Seemingly random systems have the capacity to Seemingly random systems have the capacity to generate structured behaviourgenerate structured behaviour– e.g. lots of ants that are similar, communicate locally, yet e.g. lots of ants that are similar, communicate locally, yet

develop complex communities – and these look surprisingly develop complex communities – and these look surprisingly alikealike

– an ECG pattern from different hearts is similar yet is the an ECG pattern from different hearts is similar yet is the result of many interacting cells - a ‘complex’ rhythm result of many interacting cells - a ‘complex’ rhythm developsdevelops

• The interaction between the ‘units’, each other and The interaction between the ‘units’, each other and their environment is creating new informationtheir environment is creating new information

• This ‘emergent’ behaviour can now be quantified This ‘emergent’ behaviour can now be quantified and analysed : for our purposes, we can quantify the and analysed : for our purposes, we can quantify the emergence of structure in a protein from dynamicsemergence of structure in a protein from dynamics

Complex systems: the property of Complex systems: the property of emergenceemergence

Examples of emergence in Examples of emergence in simple systemssimple systems• The software package StarLogo (from theThe software package StarLogo (from the Media Laboratory, MIT, Cambridge, Media Laboratory, MIT, Cambridge,

Massachusetts, free) allows simple systems of self organisation to be constructed.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 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 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 to move away from the other turtles. With these two rules, the turtles arrange themselves into a circle.themselves into a circle.

• 2. This project explores a simple ecosystem made up of rabbits and grass. The 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 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 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 energy, it reproduces. If it doesn't gain enough energy, it dies. The population of rabbits and grass develops an ‘attractor’ – predator/prey balance.rabbits and grass develops an ‘attractor’ – predator/prey balance.

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 systemsPart 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)

How to quantify complexityHow to quantify complexity• We have used a method called We have used a method called

computational mechanics (Crutchfield):computational mechanics (Crutchfield):• An Information based method using An Information based method using

computation theory (‘Shannon entropy and computation theory (‘Shannon entropy and Kolmogorov complexity’)Kolmogorov complexity’)

• Discovers dynamical patternsDiscovers dynamical patterns• Emergence is computed from the ability of Emergence is computed from the ability of

the system to process information – which the system to process information – which means….means….

J. P. Crutchfield and K. Young, Phys. Rev. Lett., 63, 105 (1989)J. P. Crutchfield and K. Young, Phys. Rev. Lett., 63, 105 (1989)

What does it really do ?What does it really do ?• When analysing a trajectory from the past to the When analysing a trajectory from the past to the

future, patterns emerge. To quantify the information future, patterns emerge. To quantify the information in the system, we work out how complex a model is in the system, we work out how complex a model is required given the past trajectory, to predict future required given the past trajectory, to predict future trajectories – the memory of the system – bigger trajectories – the memory of the system – bigger memory, bigger complexitymemory, bigger complexity

• This can be quantified by something called an This can be quantified by something called an --machinemachine– Below, are two Below, are two -machines from transitions observed as a -machines from transitions observed as a

small peptide undergoes a conformational transition – one small peptide undergoes a conformational transition – one analysis of the dynamics is obviously more ‘complex’ than analysis of the dynamics is obviously more ‘complex’ than the otherthe other

How do we calculate complexity of real How do we calculate complexity of real systems ?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 sequenceThe 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(Li) is a probability of the i equivalence class

K

iii LPLPC

12log (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

2logl li ii

C P x P x

Why use complexity?Why use complexity?• You are probably familiar with You are probably familiar with

a free energy funnel – this is a free energy funnel – this is misleading – its not 2D!misleading – its not 2D!

• The complexity approach is The complexity approach is designed to quantify the self-designed to quantify the self-organisation in dynamic organisation in dynamic systems which is multi-systems which is multi-dimensional – the funnel is dimensional – the funnel is also multi-dimensional and also multi-dimensional and represents not so much an represents not so much an energy funnel, but a energy funnel, but a description of transitions that description of transitions that utilise quasi-periodic dynamicsutilise quasi-periodic dynamics

Theodore L. Brown,

Making Truth. The Roles of Metaphor in Science

Calculating the complexity ofCalculating the complexity of real systems real systems

How do we make sense out of this:

oxygen atom motions from water close to the protein trajectory

Trajectories, dynamicsTrajectories, dynamics• This is a cartoon of the This is a cartoon of the

global systemglobal system• In reality, it’s a High-In reality, it’s a High-

dimensional phase spacedimensional phase space• Our expectation was that Our expectation was that

in transitioning from one in transitioning from one state to another, state to another, complexity would changecomplexity would change

• Here we are talking Here we are talking about the change in the about the change in the dynamic behaviour dynamic behaviour (could be the protein or (could be the protein or solvent for example) of solvent for example) of the systemthe system

• Like two guitar strings – E Like two guitar strings – E moves differently from A.moves differently from A.

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

2 waters2 waters

Initial conditions – chaotic motionEvolve into quasi-periodic motion

This is an atractor, a state into whichthe dynamic system approaches having a single basin of attraction

2 waters2 waters

• 6 degrees of freedom (for one molecule)6 degrees of freedom (for one molecule)

10 20 30 40 50 60

1.021.041.061.08

time, ps

0.98

1.00

1.02

1.04

1.08

1.10

1.12

1.14

0

2

4

0

2

4

0

2

4

OxygenCoordinates

Orientationof Dipole

Change in dynamic regime

2 waters2 waters

10 20 30 40 50 602.5

3.0

3.5

4.0

4.5

5.0

5.5 hydrogen 1 hydrogen 2

stat

istic

al c

ompl

exity

time, ps

0

2

40

1

2

3

0

2

4

Orientationof Dipole

Notice the change in Complexity oftwo hydrogen atoms of a water moleculeas the simulation progresses and goes from chaotic to quasi-periodic motion

Trajectories, dynamicsTrajectories, dynamics• Back to a cartoon Back to a cartoon

of the global of the global systemsystem

• Do the transitions Do the transitions happen in a similar happen in a similar ‘concerted’ ‘concerted’ fashion, with a fashion, with a decrease in decrease in complexity of complexity of dynamics?dynamics?

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

(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-enkephalinX-ray structure

-turn form

How does complexity change for the peptide and for the water?

open form

enkephalinenkephalin

Formation of a -turn

An example of peptide Dynamic An example of peptide Dynamic regimesregimes

-1000

-500

0

500

1000

0 5000 10000 15000 20000

tim e, ps

dih

edra

l an

gle,

deg

rees

An example of atransition (fast)from one minimum toanother for this dihedral angle

Complexity analysis of peptide/water molecules during the folding Complexity analysis of peptide/water molecules during the folding eventevent

turnevent turn

eventTopological 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

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

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 ?

Sensitivity of folding transitions Sensitivity of folding transitions to small perturbations of the to small perturbations of the solvent or peptidesolvent or peptide• A larger system. Chignolin, 10 amino A larger system. Chignolin, 10 amino

acidsacidsGLY-TYR-ASP-PRO-GLU-THR-GLY-THR-TRP-GLY

Simulated for 1ns and transitions in conformation analysed.

chignolin, showing a chignolin, showing a -turn-turn

nmr-structure 1UAO

Chignolin – showing how a small Chignolin – showing how a small perturbation (lower) to the velocity of perturbation (lower) to the velocity of one atom prevents a transition (upper)one atom prevents a transition (upper)

-15 0

-10 0

-50

0

5 0

1 00

1 50

1 31 10 1 31 15 1 31 20 1 31 25 1 31 30 1 31 35 1 31 40 1 31 45 1 31 50

-15 0

-10 0

-50

0

5 0

1 00

1 50

1 31 10 1 31 15 1 31 20 1 31 25 1 31 30 1 31 35 1 31 40 1 31 45 1 31 50

Perturbation appliedhere – transition doesn’thappen!This can be applied to a watermolecule 15Ǻ away from the peptide!So, the ‘tunnel’ is very tight. The whole system is in concerted motion at this time

ConclusionsConclusions

• ‘‘Why’ proteins fold can be viewed as a Why’ proteins fold can be viewed as a phenomenon of the dynamics of the systemphenomenon of the dynamics of the system

• Complexity analysis can provide a different Complexity analysis can provide a different perspectiveperspective

• Folding transitions follow very tight concerted Folding transitions follow very tight concerted motionsmotions

• Small perturbations can disrupt transitions – even Small perturbations can disrupt transitions – even up to 15up to 15ǺǺ away. away.

• Caveat: all the dynamics simulations were Caveat: all the dynamics simulations were performed using Gromacs and the results are performed using Gromacs and the results are obviously subject to how accurately the obviously subject to how accurately the methodology reflects protein dynamics.methodology reflects protein dynamics.

AcknowledgementsAcknowledgements

• George Karvounis, Herman Berendsen, George Karvounis, Herman Berendsen, Makoto TaijiMakoto 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 H. J. C. Berendsen, D. van der Spoel and R. van Drunen, GROMACS: A Message-passing Parallel Molecular Dynamics Implementation, Message-passing Parallel Molecular Dynamics Implementation, Comp. Phys. Comp. Phys. CommunCommun. . , , 9191, 43-56 (1995) GROMOS W. R. P. Scott, P. H. Hunenberger , I. G. , 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, 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 W. F. van Gunsteren, The GROMOS Biomolecular Simulation Program Package, J. Phys. Chem. A, 103, 3596-3607 (1999) Package, J. Phys. Chem. A, 103, 3596-3607 (1999)

212

0

400

EC

G

time

200 250

-400

0

400

800

EC

G

time

0 20000 40000 600005

6

7

8

9

com

plex

ity

time steps

ECG data in collaboration with Addenbrookes hospital Dr IB WilkinsonClinical Pharmacology Unit, Addenbrooke's Hospital, Cambridge

Complexity for ‘normal’heart rhythm

Change in complexityupon drug dosing

Moving the subject

drugged heart (continuation of red curve)