3. Optimization Methods for Molecular Modeling by Barak Raveh.

3. Optimization Methods for Molecular Modeling

by Barak Raveh

Outline• Introduction• Local Minimization Methods (derivative-based)

– Gradient (first order) methods– Newton (second order) methods

• Monte-Carlo Sampling (MC)– Introduction to MC methods– Markov-chain MC methods (MCMC)– Escaping local-minima

Prerequisites for Tracing the Minimal Energy Conformation

I. The energy function:The in-silico energy function should correlate with the (intractable) physical free energy. In particular, they should share the same global energy minimum.

II. The sampling strategy:Our sampling strategy should efficiently scan the (enormous) space of protein conformations

The Problem: Find Global Minimum on a Rough One Dimensional Surface

rough = has multitude of local minima in a multitude of scales.

*Adapted from slides by Chen Kaeasar, Ben-Gurion University

The landscape is rough because both small pits and the Sea of Galilee are local

minima.

The Problem: Find Global Minimum on a Rough Two Dimensional Surface


The Problem: Find Global Minimum on a RoughMulti-Dimensional Surface

• A protein conformation is defined by the set of Cartesian atom coordinates (x,y,z) or

by Internal coordinates (φ /ψ/χ torsion angles ; bond angles ; bond lengths)

• The conformation space of a protein with 100 residues has ≈ 3000 dimensions

• The X-ray structure of a protein is a point in this space.

• A 3000-dimensional space cannot be systematically sampled, visualized or

comprehended.


Characteristics of the Protein Energetic Landscape

smooth? rugged?

Images by Ken Dill

space of conformations

energy

Example: removing clashes from X-ray models

Local Minimization Allows the Correction of Minor Local Errors in Structural Models



The path to the closest local minimum = local minimization

What kind of minima do we want?

A Little Math – Gradients and Hessians Gradients and Hessians generalize the first and second derivatives (respectively)

of multi-variate scalar functions ( = functions from vectors to scalars)

jijiji

jijiji

jijiji

ji

zz

E

yz

E

xz

E

zy

E

yy

E

xy

E

zx

E

yx

E

xx

E

rr

Eij

222

222

222

2

h

i

i

i

ii

z

Ey

Ex

E

r

E

Gradient Hessian

Energy = f(x1, y1, z1, … , xn, yn, zn)

Analytical Energy Gradient (i) Cartesian Coordinates

Energy, work and force: recall that Energy ( = work) is defined as force integrated over distance Energy gradient in Cartesian coordinates = vector of forces that act upon atoms (but this is not exactly so for statistical energy functions, that aim at the free energy ΔG)

Energy, work and force: recall that Energy ( = work) is defined as force integrated over distance Energy gradient in Cartesian coordinates = vector of forces that act upon atoms (but this is not exactly so for statistical energy functions, that aim at the free energy ΔG)

E = f(x1, y1 ,z1, … , xn, yn, zn)

nnn z

E

y

E

x

E

z

E

y

E

x

E...

111

Example: Van der-Waals energy between pairs of atoms – O(n2) pairs:

ji ijij

VdW R

B

R

AE

,612

713

612

ijijij

VdW

R

B

R

A

R

E

222 )()()( jijijiij zzyyxxR

Enrichment: Transforming a gradient between Cartesian and Internal coordinates (see Abe, Braun, Nogoti and Gö, 1984 ; Wedemeyer and Baker, 2003)

Consider an infinitesimal rotation of a vector r around a unit vector n . From physical

mechanics, it can be shown that:

Analytical Energy Gradient (ii) Internal Coordinates (torsions, etc.)

E = f(1, 1, 1, 11, 12 , …)

...1211111 EEEEE

Note: For simplicity, bond lengths and bond angles are often ignored

rnr

cross product – right hand rule

n x r

r

n

Using the fold-tree (previous lesson), we can recursively propagate changes in internal coordinates to the whole structure (see Wedemeyer and Baker 2003)

n

adapted from image by Sunil Singh http://cnx.org/content/m14014/1.9/

r

Gradient Calculations – Cartesian vs. Internal Coordinates

For some terms, Gradient computation is simpler and more natural with Cartesian coordinates, but harder for others:• Distance / Cartesian dependent: Van der-Waals term ; Electrostatics ; Solvation

• Internal-coordinates dependent: Bond length and angle ; Ramachandran and Dunbrack terms (in Rosetta)

• Combination: Hydrogen-bonds (in some force-fields)

Reminder: Internal coordinates provide a natural distinction between soft constraints (flexibility of φ/ψ torsion angles) and hard constraints with steep gradient (fixed length of covalent bonds). Energy landscape of Cartesian coordinates is more rugged.

Reminder: Internal coordinates provide a natural distinction between soft constraints (flexibility of φ/ψ torsion angles) and hard constraints with steep gradient (fixed length of covalent bonds). Energy landscape of Cartesian coordinates is more rugged.

• Analytical solutions require a closed-form algebraic formulation of energy score

• Numerical solution try to approximate the gradient (or Hessian)– Simple example:

f’(x) ≈ f(x+1) – f(x)– Another example:

the Secant method (soon)

Analytical vs. Numerical Gradient Calculations

Gradient Descent Minimization AlgorithmSliding down an energy gradient

good ( = global minimum)

local minimum

Image by Ken Dill

1. Coordinates vector (Cartesian or Internal coordinates):

X=(x1, x2,…,xn)

2. Differentiable energy function:

E(X)

3. Gradient vector:

xxx n

EEEX ,......,,)(

21


Gradient Descent – System Description

Gradient Descent Minimization Algorithm:

Parameters: λ = step size ; = convergence threshold

• x = random starting point• While (x) >

– Compute (x)– xnew = x + λ(x)

• Line search: find the best step size λ in order to minimize E(xnew) (discussion later)

Note on convergence condition: in local minima, the gradient must be zero (but not always the other way around)Note on convergence condition: in local minima, the gradient must be zero (but not always the other way around)

Line Search Methods – Solving argminλE[x + λ(x)]:

(1)This is also an optimization problem, but in one-dimension…(2)Inexact solutions are probably sufficientInterval bracketing – (e.g., golden section, parabolic interpolation, Brent’s search)

– Bracketing the local minimum by intervals of decreasing length – Always finds a local minimum

Backtracking (e.g., with Armijo / Wolfe conditions): – Multiply step-size λ by c<1, until some condition is met– Variations: λ can also increase

1-D Newton and Secant methodsWe will talk about this soon…

The (very common) problem: a narrow, winding “valley” in the energy landscape The narrow valley results in miniscule, zigzag steps

The (very common) problem: a narrow, winding “valley” in the energy landscape The narrow valley results in miniscule, zigzag steps

2-D Rosenbrock’s Function: a Banana Shaped ValleyPathologically Slow Convergence for Gradient Descent

100 iterations

1000 iterations

0 iterations

10 iterations

(One) Solution: Conjugate Gradient Descent• Use a (smart) linear combination of gradients from previous

iterations to prevent zigzag motion

Parameters: λ = step size ; = convergence threshold

• x0 = random starting point• Λ0 = (x0)• While Λi >

– Λi+1 = (xi) + βi∙Λi– choice of βi is important

– Xi+1 = xi + λ ∙ Λi • Line search: adjust step size λ to

minimize E(Xi+1)

gradient descent

Conjugated gradient descent

• The new gradient is “A-orthogonal” to all previous search direction, for exact line search• Works best when the surface is approximately quadratic near the minimum (convergence in N iterations),

otherwise need to reset the search every N steps (N = dimension of space)

• The new gradient is “A-orthogonal” to all previous search direction, for exact line search• Works best when the surface is approximately quadratic near the minimum (convergence in N iterations),

otherwise need to reset the search every N steps (N = dimension of space)

Root Finding – when is f(x) = 0?

Taylor’s Series

First order approximation:

Second order approximation:

The full Series:

Example:

=

(a=0)

Taylor’s Approximation: f(x)=ex

Taylor’s Approximation of f(x) = sin(x)2x at x=1.5

-2

-1

0

1

2

-3-2-10123

sin(X)^(2X)


-2

-1

0

1

2

-3-2-10123

sin(X)^(2X)

1st order


-2

-1

0

1

2

-3-2-10123

sin(X)^(2X)

1st order

2nd order

-2

-1

0

1

2

-3-2-10123

sin(X)^(2X)1st order2nd order3rd order


From Taylor’s Series to Root Finding(one-dimension)

First order approximation:

)(!1

)(')()( ax

afafxf

Root finding by Taylor’s approximation:

)(!1

)(')(0 ax

afaf

)('

)(

af

afax

Newton-Raphson Method for Root Finding(one-dimension)

1. Start from a random x0

2. While not converged, update x with Taylor’s series:

)('

)(1

n

nnn xf

xfxx

Image from http://www.codecogs.com/d-ox/maths/rootfinding/newton.php

Newton-Raphson: Quadratic Convergence Rate

THEOREM: Let xroot be a “nice” root of f(x). There exists a “neighborhood” of some size Δ around xroot , in which Newton method will converge towards xroot quadratically ( = the error decreases quadratically in each round)

The Secant Method(one-dimension)

• Just like Newton-Raphson, but approximate the derivative by drawing a secant line between two previous points:

01

01 )()()('

xx

xfxfxf

Secant algorithm:

1. Start from two random points: x0, x1

2. While not converged:

•Theoretical convergence rate: golden-ratio (~1.62)•Often faster in practice: no gradient calculations

Newton’s Method:from Root Finding to Minimization

Second order approximation of f(x):

2)(!2

)('')(

!1

)(')()( ax

afax

afafxf

Minimum is reached when derivative of approximation is zero:

))(('')('0 axafaf

)(''

)('

af

afax

take derivative (by X)

• So… this is just root finding over the derivative (which makes sense since in local minima, the gradient is zero)

• So… this is just root finding over the derivative (which makes sense since in local minima, the gradient is zero)

Newton’s Method for Minimization:

1. Start from a random vector x=x0


)(''

)('

xf

xfxxnew

Notes: • if f’’(x)>0, then x is surely a local minimum point• We can choose a different step size than one

Newton’s Method for Minimization:Higher Dimensions

1. Start from a random vector x=x0


Notes: •H is the Hessian matrix (generalization of second derivative to high dimensions)•We can choose a different step size using Line Search (see previous slides)

Generalizing the Secant Method to High Dimensions: Quasi-Newton Methods

• Calculating the Hessian (2nd derivative) is expensive numerical calculation of Hessian

• Popular methods: – DFP (Davidson – Fletcher – Powell)– BFGS (Broyden – Fletcher – Goldfarb – Shanno)– Combinations

Timeline: Newton-Raphson (17th century) Secant method DFP (1959, 1963)

Broyden Method for roots (1965) BFGS (1970)

Timeline: Newton-Raphson (17th century) Secant method DFP (1959, 1963)

Broyden Method for roots (1965) BFGS (1970)

Some more Resources on Gradient and Newton Methods

• Conjugate Gradient Descent http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf

• Quasi-Newton Methods: http://www.srl.gatech.edu/education/ME6103/Quasi-Newton.ppt

• HUJI course on non-linear optimization by Benjamin Yakir http://pluto.huji.ac.il/~msby/opt-files/optimization.html

• Line search:– http://pluto.huji.ac.il/~msby/opt-files/opt04.pdf– http://www.physiol.ox.ac.uk/Computing/Online_Documentation/Matlab/toolbox/nnet/backpr59.html

• Wikipedia…

http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf

http://www.srl.gatech.edu/education/ME6103/Quasi-Newton.ppt

http://pluto.huji.ac.il/~msby/opt-files/optimization.html

http://pluto.huji.ac.il/~msby/opt-files/opt04.pdf

http://www.physiol.ox.ac.uk/Computing/Online_Documentation/Matlab/toolbox/nnet/backpr59.html

Arbitrary starting point

Example: predict protein structure from its AA sequence.

Harder Goal: Move from an Arbitrary Model to a Correct One


10

iteration


100

iteration


200

iteration


400

iteration


800

iteration


1000

iteration


1200

iteration


1400

iteration


1600

iteration


1800

iteration


2000

iteration


4000

iteration


7000

iterationThis time succeeded, in many cases not.


What kind of paths do we want?

The path to the global minimum


Monte-Carlo Methods(a.k.a. MC simulations, MC sampling or MC search)

• Monte-Carlo methods (“casino” methods) are a very general term for estimations that are based on a series of random samples– Samples can be dependent or independent– MC physical simulations are most famous for their

role in the Manhattan Project (Uncle of Polish mathematician Stanisław Marcin Ulam’s was said to be a heavy gambler)

Example: Estimating Π by Independent Monte-Carlo Samples (I)

Suppose we throw darts randomly (and uniformly) at the square:

Algorithm:For i=[1..ntrials] x = (random# in [0..r]) y = (random# in [0..r]) distance = sqrt (x^2 + y^2) if distance ≤ r hits++EndOutput:

ntrials

hits4

Adapted from course slides by Craig Douglas

http://www.chem.unl.edu/zeng/joy/mclab/mcintro.html




Drunk Sailor’s Random Walk http://www.chem.uoa.gr/applets/AppletSailor/Appl_Sailor2.html

What is the probability that the sailor will leave through each exit?What is the probability that the sailor will leave through each exit?

0.25

0.25

0.250.25

0.25

0.250.25

0.25

http://www.chem.uoa.gr/applets/AppletSailor/Appl_Sailor2.html

http://www.chem.uoa.gr/applets/AppletSailor/Appl_Sailor2.html

Markov-Chain Monte Carlo (MCMC)

• Markov-Chain: future state depends only on present state

• Markov-Chain Monte-Carlo on Graphs: we randomly walk from node to node with a certain probability, that depends only on our current location.

0.5

0.5

0.25

0.75

Analysis of a Two-Nodes Walk

A B 0.5

0.5

0.25

0.75After n rounds, what is the probability of being in node A?

After n rounds, what is the probability of being in node A?

Assume Prn+1A ≈ PrnA for a large n:

Prn+1A = PrnA x 0.75 + PrnB x 0.5

0.25 x PrnA = PrnB x 0.5

PrnA = 2 x PrnB

So: Pr∞A = ⅔ Pr∞B = ⅓

Assume Prn+1A ≈ PrnA for a large n:

Prn+1A = PrnA x 0.75 + PrnB x 0.5

0.25 x PrnA = PrnB x 0.5

PrnA = 2 x PrnB

So: Pr∞A = ⅔ Pr∞B = ⅓

After a long run, we want to find low-energy conformations, with high probability

After a long run, we want to find low-energy conformations, with high probability

Sampling Protein Conformations with MCMC

Protein image taken from Chemical Biology, 2006

Markov-Chain Monte-Carlo (MCMC) with “proposals”:1. Perturb Structure to create a “proposal”2. Accept or reject new conformation with a “certain” probability

Markov-Chain Monte-Carlo (MCMC) with “proposals”:1. Perturb Structure to create a “proposal”2. Accept or reject new conformation with a “certain” probability

But how?But how?

A (physically) natural* choice is the Boltzmann distribution, proportional to:

Ei = energy of state ikB = Boltzmann constantT = temperatureZ = “Partition Function” constant

A (physically) natural* choice is the Boltzmann distribution, proportional to:

Ei = energy of state ikB = Boltzmann constantT = temperatureZ = “Partition Function” constant

* In theory, the Boltzmann distribution is a bit problematic in non-gas phase, but never mind that for now…* In theory, the Boltzmann distribution is a bit problematic in non-gas phase, but never mind that for now…

Ze Tk

E

B

i

The Metropolis-Hastings Criterion

• Boltzmann Distribution:

• The energy score and temperature are computed (quite) easily• The “only” problem is calculating Z (the “partition function”) –

this requires summing over all states.• Metropolis showed that MCMC will converge to the true

Boltzmann distribution, if we accept a new proposal with

probability

"Equations of State Calculations by Fast Computing Machines“ – Metropolis, N. et al. Journal of Chemical Physics (1953)

Z

e TkE Bi

If we run till infinity, with good perturbations, we will visit every conformation according to the Boltzmann distribution

If we run till infinity, with good perturbations, we will visit every conformation according to the Boltzmann distribution

Sampling Protein Conformations with Metropolis-Hastings MCMC

Protein image taken from Chemical Biology, 2006

Markov-Chain Monte-Carlo (MCMC) with “proposals”:1. Perturb Structure to create a “proposal”2. Accept or reject new conformation by the Metropolis criterion3. Repeat for many iterations

Markov-Chain Monte-Carlo (MCMC) with “proposals”:1. Perturb Structure to create a “proposal”2. Accept or reject new conformation by the Metropolis criterion3. Repeat for many iterations

But we just want to find the energy minimum. If we do our perturbations in a smart manner, we can still cover relevant (realistic, low-energy) parts of the search space

But we just want to find the energy minimum. If we do our perturbations in a smart manner, we can still cover relevant (realistic, low-energy) parts of the search space


Getting stuck in a local minimum

Trick 1: Simulated Annealing

The Boltzmann distribution depends on the in-silico temperature T:• In low temperatures, we will get stuck in local minima (we will

always get zero if the energy rises even slightly)• In high temperatures, we will always get 1 (jump between

conformations like nuts).

The Boltzmann distribution depends on the in-silico temperature T:• In low temperatures, we will get stuck in local minima (we will

always get zero if the energy rises even slightly)• In high temperatures, we will always get 1 (jump between

conformations like nuts).

In simulated annealing, we gradually decrease (“cool down”) the virtual temperature factor,

until we converge to a minimum point

In simulated annealing, we gradually decrease (“cool down”) the virtual temperature factor,

until we converge to a minimum point

Trick 2: Monte-Carlo with Energy Minimization (MCM)Scheraga et al., 1987

• Derivative-based methods (Gradient Descent, Newton’s method, DFP) are excellent at finding near-by local minima

• In Rosetta, Monte-Carlo is used for bigger jumps between near-by local minima

Trick 3: Switching between Low-Resolution (smooth) and High-Resolution (rugged) energy functions

• In Rosetta, the Centroid energy function is used to quickly sample large perturbations

• The Full-Atom energy function is used for fine tuning

START

energy

conformations

Smooth Low-res

Rugged High-res

Trick 4: Repulsive Energy Ramping• The repulsive VdW energy is the main reason for getting stuck• Start simulations with lowered repulsive energy term, and gradually ramp

it up during the simulation• Similar rational to Simulated Annealing

Trick 5: Modulating Perturbation Step Size• A too small perturbation size can lead to a very slow simulation

we remain stuck in the local minimum• A large perturbation size can lead to clashes and a very high rejection rate

we remain stuck in the same local minimum• We can increase or decrease the step size until a fixed rejection rate (for

example, 50%) is achieved

Monte-Carlo in Rosetta

• In Rosetta, it is common to use any of the above tricks, MCM in particular

• In general, a single simulation is pretty short (no more than a few minutes), but is repeated k independent times – getting k sampled “decoys” – We use energy scoring to decide which is the best decoy structure –

hopefully this is the near-native solution– Low-resolution sampling is often used to create a very large number

of initial decoys, and only the best ones are moved to high-resolution minimization

Summary• Derivative-based methods can effectively reach near-by energy minima

• Metropolis-Hastings MCMC can recover the Boltzmann distribution in some applications, but for protein folding, we cannot hope to cover the huge conformational space, or recover the Boltzmann distribution.

• Still, useful tricks help us find good low-energy near-native conformations (Simulated Annealing, Monte-Carlo with Minimization, Centroid mode, Ramping, Step size modulation, and other smart sampling steps, etc.).

• We didn’t cover some very popular non-linear optimization methods:– Linear and Convex Programming ; Expectation Maximization algorithm ; Branch

and Bound algorithms ; Dead-End Elimination (Lesson 4) ; Mean Field approach ; And more…

3. Optimization Methods for Molecular Modeling by Barak Raveh.

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Transcript of 3. Optimization Methods for Molecular Modeling by Barak Raveh.