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Bayesian

statistical

analysis

of

protein

side-chain

rotamer

preferences

R.

L.

DUNBRACK-JR

and

F.

E.

COHEN

Protein

Sci

References

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Protein

Science

-LRB-

-RRB-

Cambridge

University

Press

Printed

in

the

USA

Copyright

The

Protein

Society

Bayesian

statistical

analysis

of

protein

side-chain

rotamer

preferences

ROLAND

L.

DUNBRACK

JR.

AND

FRED

E.

COHEN

Department

of

Cellular

and

Molecular

Pharmacology

University

of

California

San

Francisco

San

Francisco

California

-LRB-

RECEIVED

December

ACCEPTED

April

-RRB-

Abstract

We

present

a

Bayesian

statistical

analysis

of

the

conformations

of

side

chainsin

proteins

from

the

Protein

Data

Bank

This

is

an

extension

of

the

backbone-dependent

rotamer

library

and

includes

rotamer

populations

and

average

y

angles

for

a

full

range

of

values

The

Bayesian

analysis

used

here

provides

a

rigorous

statistical

method

for

taking

account

of

varying

amounts

of

data

Bayesian

statistics

requires

the

assumption

of

aprior

distribution

for

parameters

over

their

range

of

possible

values

This

prior

distribution

can

be

derived

from

previous

data

or

from

pooling

some

of

the

present

data

The

prior

distribution

is

combined

with

the

data

to

form

the

posterior

distribution

which

is

a

compromise

between

the

prior

distribution

and

the

data

For

the

y

and

rotamer

prior

distributions

we

assume

that

the

probability

of

each

rotamer

type

is

dependent

only

on

the

previous

y

rotamer

in

the

chain

For

the

backbone-dependence

of

the

y

I

rotamers

we

derive

prior

distributions

from

the

product

of

the

dependent

and

dependent

probabilities

Molecular

mechanics

calculations

with

the

potential

show

a

strong

similarity

with

the

experimental

distributions

indicating

that

proteins

attain

their

lowest

energy

rotamers

with

respect

to

local

backbone,side-chain

interactions

The

new

library

is

suitable

for

use

in

homology

modeling

protein

folding

simulations

and

the

refinement

of

X-ray

and

NMR

structures

Keywords

Bayesian

statistics

molecular

mechanics

protein

structure

rotamers

side

chains

In

the

last

I

O

years

the

Protein

Data

Bank

-LRB-

PDB

-RRB-

has

more

than

tripled

in

size

As

the

PDB

has

grown

it

has

become

possible

to

analyze

protein

structure

in

greater

detail

and

with

greater

statistical

certainty

Although

some

studies

have

been

qualitative

in

nature

the

size

of

the

database

now

enables

us

to

put

some

structural

variables

on

a

more

sound

statistical

footing

One

aspect

of

protein

structure

that

has

been

analyzed

over

a

period

of

many

years

is

the

distribution

of

protein

side-chain

conformations

Sidechain

rotamer

libraries

-LRB-

Chandrasekaran

Ramachandran

Cody

et

al.

James

Sielecki

Ponder

Richards

-RRB-

consist

of

a

list

of

discrete

side-chain

conformations

and

their

associated

probabilities

determined

from

their

frequency

of

occurrence

in

the

PDB

In

most

cases

these

conformations

correspond

to

rotamers

or

local

minima

on

potential

energy

maps

-LRB-

Bhat

et

al.

Gelin

Karplus

Benedetti

et

al.

-RRB-

with

frequencies

predictable

from

conformational

analysis

of

organic

molecules

-LRB-

Janin

et

al.

Dunbrack

Karplus

-RRB-

The

discreteness

of

rotamers

is

enforced

by

barriers

of

kcal

mol

due

to

the

overlap

of

bond

molecular

orbitals

in

eclipsed

conformations

-LRB-

Karplus

Parr

-RRB-

Reprint

requests

to

Roland

L.

Dunbrack

Jr.

Department

of

Cellular

and

Molecular

Pharmacology

University

of

California

San

Francisco

San

Francisco

California

e-mail

dunbrack@cmpharm.ucsf.edu

Address

after

October

Institute

for

Cancer

Research

Fox

Chase

Cancer

Center

Burholme

Avenue

Philadelphia

PA

As

more

high-resolution

structures

have

become

available

in

recent

years

it

has

become

possible

to

determine

rotamer

preferences

asa

function

of

backbone

conformation

Earlier

efforts

found

weak

correlations

of

rotamer

distributions

in

different

secondary

structures

-LRB-

Janin

et

al.

McGregor

et

al.

Sutcliffe

et

al.

Schrauber

et

al.

-RRB-

With

an

extended

database

of

protein

chains

with

resolution

better

than

or

equal

to

we

compiled

a

backbone-dependent

rotamer

library

-LRB-

Dunbrack

Karplus

-RRB-

that

gives

the

side-chain

y

I

rotamer

distribution

for

each

amino

acid

type

and

each

occupied

by

region

of

the

conformation

space

of

the

backbone

y

l

rotamer

preferences

show

detectable

patterns

as

a

function

of

and

for

all

side

chains

that

can

be

explained

by

simple

steric

conformational

analysis

-LRB-

Dunbrack

Karplus

-RRB-

The

backbone-dependent

rotamer

library

has

been

shown

to

be

useful

as

a

tool

for

predicting

side-chain

conformations

from

backbone

coordinates

for

homology

modeling

-LRB-

Dunbrack

Karplus

Bower

et

al.

-RRB-

and

for

NMR

and

X-ray

structure

refinement

-LRB-

Kuszewski

et

al.

-RRB-

The

results

obtained

are

a

significant

improvement

over

backbone-independent

rotamer

libraries

Although

the

larger

database

presently

available

affords

a

more

complete

view

of

the

variation

of

rotamer

preferences

as

a

function

of

and

much

of

the

Ramachandran

map

is

sparsely

populated

because

of

backbone-backbone

steric

exclusions

Even

in

wellpopulated

regions

of

the

Ramachandran

map

some

rotamers

are

quite

rare

and

their

frequencies

are

therefore

statistically

unreli

Downloaded

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Cold

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Laboratory

Press

able

For

the

purposes

of

homology

modeling

protein

folding

simulations

experimental

structure

refinement

and

comparison

with

energy

calculations

we

need

more

complete

estimates

of

rotamer

distributions

than

are

available

in

the

database

In

this

paper

we

use

Bayesian

statistical

analysis

to

account

for

the

varying

amount

of

informationinthedatabase

for

I

backboney

dependent

rotamer

distributions

In

addition

certain

combinations

of

the

x

I

y

and

rotamers

are

rare

and

the

Bayesian

analysis

provides

a

better

estimate

of

their

probability

of

occurrence

than

do

the

data

alone

Such

sparse

data

corrections

have

been

used

in

the

calculations

of

potentials

of

mean

force

from

PDB

data

by

Sippl

-LRB-

-RRB-

and

by

Sali

and

Blundell

-LRB-

-RRB-

although

not

labeled

Bayesian

statistics

as

such

Briefly

Bayesian

statistical

analysis

provides

a

framework

for

combining

prior

information

about

measurable

quantities

with

-LRB-

usually

limited

-RRB-

experimental

data

to

determine

a

better

estimate

of

a

parameter

of

interest

than

the

data

alone

provide

A

simple

example

is

in

the

flipping

of

a

coin

that

may

be

slightly

biased

As

prior

information

perhaps

after

a

visual

inspection

we

might

posis

fairly

close

to

We

tulate

that

the

probability

of

heads

-LRB-

e

-RRB-

guess

a

probability

distribution

for

Ohrod

that

extends

between

and

but

is

peaked

around

ehpod

Ifwe

flip

the

coin

times

and

get

heads

we

would

not

be

justified

in

saying

that

ehuod.

is

We

need

to

consider

the

likelihood

of

getting

heads

from

I

O

tosses

given

all

possible

values

for

the

underlying

parameter

Ohradr

The

likelihood

function

is

denoted

p

-LRB-

l

-RRB-

because

it

is

the

probability

of

the

data

y

given

the

parameter

In

Figure

we

show

a

prior

distribution

and

likelihood

function

for

the

coin

flipping

example

It

is

clear

that

the

likelihood

of

getting

heads

from

tosses

is

significant

even

when

eh

The

goal

of

Bayesian

analysis

is

a

full

view

of

the

distribution

of

Ohpad

given

the

data

in

the

coin

flipping

example

p

-LRB-

Oht

ad

y

n

-RRB-

central

equation

of

Bayesian

statistics

deI

The

R.L.

Dunbrack

JI

and

RE

Cohen

scribed

in

Materials

and

methods

combines

the

prior

distribution

and

the

likelihood

function

to

reach

the

posterior

distribution

P

-LRB-

OlY

-RRB-

P

-LRB-

YI

-RRB-

P

-LRB-

-RRB-

-LRB-

-RRB-

I

I

I

I

I

prior

data

h

-LRB-

d

v

a

As

shown

in

Figure

in

the

coin

flipping

example

the

posterior

distribution

for

y

peaks

just

above

because

of

the

observed

data

of

heads

from

tosses

The

extent

of

the

shift

from

the

data

value

-LRB-

-RRB-

isincorporated

into

the

analysis

by

the

form

of

the

prior

distribution

In

any

case

as

the

number

of

observations

n

increases

the

resulting

distribution

p

-LRB-

Bhead

rl

yhead

y

n

-RRB-

becomes

more

concentrated

at

the

observed

ratio

of

heads

to

tosses

It

is

central

to

Bayesian

analysis

that

the

posterior

distribution

is

more

than

a

point

estimate

for

a

parameter

such

as

e

k

e

o

instead

it

is

a

probability

distribution

over

the

full

range

of

allowed

values

of

the

parameter

This

aspect

can

be

exploited

by

simulation

where

values

for

the

parameter

can

be

drawn

randomly

from

the

posterior

distribution

and

used

to

explore

the

distribution

of

any

function

of

the

parameter

This

method

is

used

here

to

show

how

well

the

calculated

posterior

distributions

for

rotamer

probabilities

correspond

to

the

raw

data

in

the

PDB

With

a

full

analysis

of

the

rotamer

preferences

of

protein

side

chains

from

Bayesian

statistics

we

compare

the

experimental

data

with

molecular

mechanics

calculations

using

the

C

H

A

R

potential

Previous

calculations

-LRB-

Ponnuswamy

Sasisekharan

Sasisekharan

Ponnuswamy

Pullman

Pullman

Janin

et

al.

Marcus

et

al.

-RRB-

emphasized

the

limitations

that

individual

side

chains

place

on

the

backbone

conformation

In

contrast

we

show

the

potential

energy

surfaces

in

a

way

that

captures

the

limitations

the

backbone

places

on

the

side-chain

conformation

and

we

compare

the

results

with

conformational

analysis

based

on

data

on

hydrocarbons

such

as

butane

and

pentane

as

described

previously

-LRB-

Dunbrack

Karplus

-RRB-

and

with

the

experimental

distributions

derived

from

the

Bayesian

analysis

A

number

of

protein-folding

models

suggests

that

backbone

conformations

are

achieved

earlier

in

folding

than

final

side-chain

conformations

-LRB-

Shakhnovich

Finkelstein

Bromberg

Dill

Dill

et

al.

-RRB-

If

this

is

the

case

then

the

restrictions

that

the

backbone

places

on

side-chain

conformations

are

of

great

importance

in

understanding

the

thermodynamics

and

kinetics

of

protein

folding

In

a

subsequent

paper

we

will

examine

the

correspondence

between

energy

calculations

and

the

experimental

data

in

a

more

quantitative

fashion

with

a

statistical

mechanical

model

that

examines

the

influence

of

both

local

backbone

conformation

and

tertiary

packing

interactions

on

protein

side

chains

n

Results

Bayesian

analysis

of

r,-conditional

backbone-independent

rotamer

populations

I

I

o

theta

Fig.

Bayesian

analysis

of

a

set

of

coin

flips

The

prior

density

was

calculated

assuming

heads

from

tosses

for

a

perfect

coin

-LRB-

Ohuad

-RRB-

The

likelihood

or

data

density

was

calculated

assuming

heads

from

I

O

tosses

The

resulting

posterior

density

is

also

plotted

To

make

a

clear

distinction

between

the

y

angles

and

their

correy

sponding

rotamers

we

denote

the

I

rotamer

as

rl

the

as

etc.

The

rotamer

definitions

for

all

side

chains

and

y

angles

are

given

in

Table

Given

the

strong

dependence

of

the

yI

rotamer

probabilities

on

the

backbone

dihedrals

I$

and

we

present

the

backbone

-RRB-

I

independent

rotamer

library

as

the

probabilities

of

y

x

rotamers

conditional

on

the

yI

rotamer

The

conditional

backboneindependentrotamerlibrarythereforeconsists

of

probability

Downloaded

from

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by

Cold

Spring

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Laboratory

Press

Bayesian

statistics

o

protein

side

chains

f

value

of

means

that

the

prior

sample

size

is

equal

to

the

data

sample

size

and

the

two

parts

then

make

equal

contributions

to

the

posterior

distribution

We

find

a

value

of

K

to

be

reasonable

as

described

in

the

Materials

and

methods

In

Table

the

backbone-independent

prior

distribution

parameters

derived

from

the

raw

data

are

listed

for

several

side-chain

types

These

include

values

for

p

-LRB-

rblr

-RRB-

where

-LCB-

a

b

-RCB-

-LCB-

-RRB-

-LCB-

-RRB-

or

-LCB-

-RRB-

as

well

as

the

y

angle

averages

and

their

standard

The

values

for

these

parameters

for

the

remaining

deviations

-LRB-

T.

side

chains

are

available

on

our

website

-LRB-

Dunbrack

-RRB-

The

parameters

from

the

posterior

distribution

for

Met

are

listed

in

rl

Table

Met

has

rotamers

and

of

these

have

been

seen

less

than

times

each

in

our

set

of

protein

chains

from

the

PDB

The

posterior

distribution

provides

a

better

estimate

for

the

frequency

of

these

rare

rotamers

than

the

data

alone

because

it

contains

very

good

estimates

in

the

prior

distribution

This

isalso

the

case

for

Arg

and

Lys

with

backbone-independent

rotamers

each

many

of

which

have

never

been

seen

in

protein

structures

because

of

their

significant

steric

strain

It

is

the

case

that

for

all

side

chains

except

Asn

and

Asp

there

is

little

or

no

dependence

and

The

backbone-independent

liof

Ojli

or

X

-LRB-

j

l

i

-RRB-

on

and

brary

therefore

consists

of

values

for

the

parameters

-LCB-

X

-RCB-

-LRB-

j

k

ll

i

-RRB-

full

backbone-independent

library

is

also

The

available

on

our

website

-LRB-

Dunbrack

-RRB-

Bayesian

analysis

o

backbone-dependent

f

rotamer

populations

Determination

of

the

backbone-dependent

rotamer

library

is

more

difficult

than

the

backbone-independent

library

because

of

the

large

number

of

parameters

to

be

estimated

We

would

like

to

define

the

probabilities

of

the

three

rl

rotamers

-LRB-

two

for

Pro

-RRB-

for

all

values

of

and

on

a

grid

with

spacing

The

infinite

data

parameters

will

be

denoted

Orlab

for

the

proportion

of

side

chains

with

rl

i

in

a

region

near

-LCB-

IJ

b

-RRB-

Because

of

the

constraint

that

CiOilab

the

total

number

of

parameters

is

X

X

or

for

each

side-chain

type

Because

the

data

are

very

concenregions

-LRB-

a-helix

and

P-sheet

regions

-RRB-

we

trated

in

particular

need

some

procedure

for

estimating

the

probabilities

across

the

Ramachandran

map

The

full

procedure

is

described

in

Materials

and

methods

but

the

results

of

each

stage

will

be

illustrated

here

The

first

step

is

to

count

the

side

chains

of

each

rotamer

type

using

weighting

functions

shown

in

Figure

-LRB-

illustrated

for

a

b

-RRB-

The

weighting

function

in

Figure

defined

in

Equation

in

Materials

and

methods

has

a

Gaussian

shaped

peak

at

It

also

has

much

smaller

peaks

at

-LSB-

a

b

-RRB-

and

-LCB-

b

-RRB-

and

a

very

small

peak

at

-LSB-

b

-RRB-

Because

the

backbone-conformation

dependent

steric

interand

actions

that

affect

rotamer

preferences

are

periodic

in

every

-LRB-

Dunbrack

Karplus

-RRB-

-LRB-

described

in

detail

below

-RRB-

the

secondary

peaks

in

theweighting

function

serve

to

supplement

very

sparse

regions

with

data

from

regions

of

the

and/or

In

most

cases

Ramachandran

map

away

in

heavily

populated

regions

in

the

primary

peak

correspond

to

sparsely

populated

regions

in

the

secondary

peaks

and

vice

versa

In

Figure

we

show

the

function

used

in

our

previous

work

-LRB-

Dunbrack

Karplus

-RRB-

denoted

W

I

-LRB-

-RRB-

which

gives

weights

point

of

to

side

chains

with

and

within

of

the

and

to

allothers

In

Figure

the

product

of

the

functions

in

Figure

and

B

is

shown

This

function

denoted

W

r-Pe

-LRB-

-RRB-

counts

only

side

chainswithin

of

but

with

Table

Limits

for

rotamer

library

x

angles

rl

rotamers

of

all

residue

types

except

Pro

rotamers

of

Arg

Gln

Glu

Ile

Leu

Lys

Met

rotamers

of

Arg

Lys

Met

rotamers

of

Arg

Lys

rl

Conformation

g

x

range

t

OO-t

rotamers

of

Asn

Asp

Conformation

g

rotamers

of

Gln

Glu

rotarners

of

Phe

Tyr

His

x

range

t

g

rotamers

of

Trp

Conformation

g

t

x

range

Conformation

g

x

range

rl

rotamers

of

Pro

rl

L

t

g

Conformation

g

exo

x

range

distributions

of

p

-LRB-

-RRB-

is

in

contrast

to

traditional

This

backbone-independent

rotamer

libraries

which

consist

of

values

-LRB-

e.g.

for

p

-LRB-

r

l

r

r

z

r

r

r

-RRB-

Ponder

Richards

-RRB-

For

methionine

for

instance

we

would

like

to

estimate

values

for

the

conditional

probabilities

of

rz

j

k

given

that

rl

i

or

p

-LRB-

j

r

klr

i

-RRB-

that

would

be

measured

from

an

infinitely

large

data

set

We

denote

the

infinite

data

set

parameters

for

these

quantities

as

Olkl

We

derive

prior

distributions

from

a

supposition

that

p

-LRB-

r

z

j

r

k

l

r

i

-RRB-

a

p

-LRB-

r

j

-RRB-

p

-LRB-

r

z

j

l

r

j

-RRB-

-LRB-

-RRB-

That

is

rotamer

probabilities

along

the

chain

of

the

side

chain

are

dependent

only

on

the

previous

dihedral

We

use

the

probabilities

in

Equation

as

the

modes

-LRB-

maximum

values

-RRB-

of

prior

distributions

for

each

of

the

nine

Ojkli

for

each

rI

rotamer

i

The

factors

on

the

right-hand

side

of

Equation

come

directly

from

the

raw

data

population

probabilities

for

each

side-chain

type

Equation

is

quite

a

good

approximation

with

a

correlation

coefficient

of

for

the

raw

data

probabilities

versus

the

prior

distribution

estimate

for

Arg

Met

Gln

Glu

and

Lys

The

form

of

both

the

prior

and

posterior

distributions

is

a

Dirichlet

function

a

generalization

of

the

multinomial

distribution

In

forming

the

posterior

distribution

from

the

prior

distribution

and

the

data

we

need

to

choose

how

heavily

to

weight

the

prior

distribution

in

relation

to

the

data

An

equivalent

statement

is

that

we

need

to

define

the

variance

of

the

prior

distribution

compared

to

the

variance

of

the

data

To

accomplish

this

we

scale

the

prior

distribution

to

correspond

to

some

proportion

of

the

data

sample

size

with

a

scale

parameter

K

A

e

Downloaded

from

on

May

Published

by

Cold

Spring

Harbor

Laboratory

Press

Table

Backbone-independent

prior

distribution

parametersa

ro

R.L.

Dunbrack

Jz

and

EE

Cohen

rb

n

-LRB-

rblra

-RRB-

P

-LRB-

rblro

-RRB-

Xa

Ob

U

Xb

I

229 393 200

I

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

I

I

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

I

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

I

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

I

Leu

rz

I

rl

I

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

I

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

I

I

Phe

15 96 245 107

I

O

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

I

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

Met

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

22 754 508

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

I

I

I

55 27 542 313

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

continued

-RRB-

Downloaded

from

on

May

Published

by

Cold

Spring

Harbor

Laboratory

Press

Bayesian

statistics

of

protein

side

chains

Table

Continued

p

-LRB-

rbIry

-RRB-

I

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

523 121 222

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

I

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

I

316 218 352

IO

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

I

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

aRotamer

pairs

with

syn-pentane

-LRB-

or

-RRB-

interactions

are

in

italics

All

others

are

in

bold

Rotamer

designations

are

defined

in

Table

I

a

weight

similar

to

that

in

Figure

It

will

be

used

to

count

data

for

the

likelihood

function

-LRB-

see

below

-RRB-

To

check

that

the

periodic

and

nonperiodic

functions

do

not

alter

the

probabilities

significantly

from

our

previous

box

function

we

calculated

the

correlation

coefficients

for

values

of

-LSB-

-RRB-

with

more

than

side

chains

counted

with

Wbox.For

Wper

the

correlation

coefficient

with

W

br

x

was

and

for

W

On-per

the

correlation

coefficient

with

Wb

was

Between

W

p

e

r

and

Wnon-per

correlation

the

coefficient

was

The

next

step

is

to

use

the

weighted

counts

of

Wperto

derive

a

prior

density

distribution

for

use

in

a

Bayesian

analysis

for

the

backbone-dependent

rotamer

library

We

choose

a

prior

density

distribution

centered

on

a

product

of

dependent

densities

and

dependent

densities

i.e.

The

factors

on

the

right-hand

side

of

Equation

can

be

calculated

by

taking

the

logs

of

each

side

of

Equation

and

solving

the

resulting

linear

equations

by

singular

value

decomposition

The

approximation

in

Equation

is

not

quite

as

good

as

thebackboneindependent

case

but

good

enough

to

provide

reasonable

estimates

of

the

probabilities

throughout

the

Ramachandran

map

For

Arg

for

instance

the

correlation

coefficient

for

-LCB-

-RRB-

values

in

occupied

regions

of

the

Ramachandran

map

-LRB-

side

chains

-RRB-

between

the

prior

distribution

values

calculated

from

the

right-hand

side

of

Equation

and

the

raw

data

values

calculated

with

function

w

n

o

n

p

e

r

IS

As

with

the

backbone-independent

prior

distributions

-LRB-

Equation

-RRB-

values

of

p

-LRB-

i

l

b

-RRB-

denoted

defined

in

Equation

indicate

the

central

values

-LRB-

modes

-RRB-

of

Dirichlet

probability

distributions

or

p

-LRB-

j

l

a

b

-RRB-

In

Figure

we

show

the

prior

distribution

modes

calculated

with

the

weighting

function

Wperfor

several

side-chain

types

plotted

on

top

of

bar

charts

of

the

data

measured

with

the

nonperiodic

weighting

function

Wnon-per

for

values

of

and

h

with

y

or

Y

b

The

procedure

used

to

derive

the

curves

in

Figure

attempts

to

decouple

the

and

dependence

and

so

for

instance

the

dependence

of

the

r

rotamer

prior

density

is

much

flatter

than

the

data

density

-LRB-

the

bars

in

Figure

-RRB-

In

this

case

the

prior

density

does

not

respond

to

the

strong

variations

of

the

probabilities

of

the

other

two

rotamers

with

In

Figure

we

show

the

prior

distribution

the

data

density

calculated

with

the

weighting

function

Wper

and

with

W

n

o

n

p

e

r

and

the

posterior

distribution

for

the

three

rl

rotamers

of

Arg

Comparing

the

data

in

Wno

per

with

the

other

plots

demonstrates

t

-LRB-

r

lL

Downloaded

from

on

May

Published

by

Cold

Spring

Harbor

Laboratory

Press

Table

Methionine

backbone-independent

posterior

distribution

parameters

rl

R.L.

Dunbrack

Jr.

and

EE

Cohen

nnr

r

r

d

-LRB-

-LRB-

r

-RRB-

176 176 176

p

-LRB-

rI

rz

-RRB-

-LRB-

u

-RRB-

p

-LRB-

rz

-RRB-

-LRB-

u

-RRB-

Ave.

yI

a

-LRB-

yI

-RRB-

Ave.

yz

u

-LRB-

yn

-RRB-

Ave.

u

-LRB-

-RRB-

I

-LRB-

I

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

I

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

I

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

I

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

I

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

608 608 608

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

608 608 608

I

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

I

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

I

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

I

313 193 248

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

I

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

-RRB-

the

utility

of

the

Bayesian

method

for

combining

an

informative

prior

distribution

with

the

W

n

o

n

p

e

r

data

to

produce

the

posterior

distribution

The

W

non-per

data

cover

a

small

percentage

of

the

total

plot

with

isolated

regions

with

existent

data

and

large

regions

with

no

data

The

prior

distribution

is

a

smoothed

and

partially

symmetrized

reconstruction

of

the

data

Adding

back

the

nonperiodic

data

to

the

prior

distribution

to

produce

the

posterior

distribution

ensures

that

in

populated

regions

the

posterior

distribution

represents

the

data

quite

closely

In

less

populated

regions

however

the

prior

distribution

is

more

important

and

more

likely

to

be

accurate

than

the

nonperiodic

data

alone

In

unpopulated

regions

the

raw

data

areof

no

use

and

we

must

rely

on

the

accuracy

of

the

prior

distribution

defined

in

Equation

In

Figure

we

show

the

results

for

three

more

side-chain

types

-LRB-

Asp

Phe

and

Val

-RRB-

to

demonstrate

the

similarities

and

differences

between

various

types

of

residues

Side-chain

pairs

with

similar

stereochemistries

behave

similarly

especially

with

similar

chemistry

at

both

the

p

and

y

positions

-LRB-

e.g.

Asp-Asn

LysArg

Phe-Tyr

-RRB-

Thedifferences

between

theP-unbranched

and

branched

side-chain

distributions

are

clearly

due

to

Cylbackbone

interactions

Branching

at

y

and

electrostatic

interactions

also

affect

the

distributions

in

less

drastic

ways

-LRB-

e.g.

Asp-Arg

Lys-Phe

-RRB-

As

noted

earlier

the

backbone-dependent

library

does

not

include

a

dependence

of

rotamers

on

thebackbone

dihedrals

and

except

in

the

case

of

Asp

and

Asn

Asp

and

Asn

distributions

are

not

described

easily

as

rotamers

because

the

distributions

are

nearly

continuous

with

only

one

mode

But

there

is

a

significant

skew

to

the

distributions

due

to

backbone-conformation-independent

Nj

and

Cj

andbackbonesyn-pentaneeffectswithbackbone

conformation-dependent

electrostatic

effects

with

backbone

atoms

To

represent

the

skewness

of

the

of

residues

i

i

and

i

distributions

we

define

rotamers

for

for

Asp

andAsn

in

Table

centered

at

O

and

With

these

definitions

we

can

plot

the

probabilities

varying

with

and

There

isa

weak

dependence

of

p

-LRB-

-RRB-

on

and

for

Lys

Met

Glu

Gln

Arg

Ile

and

Leu

-LRB-

data

not

shown

-RRB-

For

example

the

values

of

p

-LRB-

I

-RRB-

for

-LRB-

t

-RRB-

range

from

to

with

and

in

an

identical

manner

in

all

of

these

side

chains

with

similar

variations

in

the

and

rotamers

But

for

weakly

there

are

not

enough

data

to

deterpopulated

values

of

and

mine

these

values

even

with

the

periodic

weighting

function

used

to

determine

the

backbone-dependence

of

the

rl

rotamers

The

situation

for

Asp

and

Asn

is

quite

different

There

are

large

probabilities

with

changes

inbackbone

dihedrals

shifts

inp

-LRB-

-RRB-

in

some

cases

shifts

of

density

of

or

more

The

backbonedependent

rotamer

library

for

Asp

and

Asn

therefore

contains

-RRB-

whereas

for

the

other

side

chains

the

backboneues

forp

-LRB-

rl

dependent

part

of

the

library

contains

only

the

parameters

for

P

-LRB-

I

-RRB-

Downloaded

from

on

May

Published

by

Cold

Spring

Harbor

Laboratory

Press

Bayesian

statistics

of

protein

side

chains

derive

posterior

distributions

for

the

backbone-dependent

averages

of

xl

The

backbone-dependent

library

now

consists

of

values

for

Oiloh

and

-LRB-

j

I

ilc

#Ja

-RRB-

These

results

can

be

combined

with

-RRB-

-LRB-

TI

ejk

li

-LRB-

j

j

j

-RCB-

-LRB-

r

j

r

k

llrl

i

-RRB-

to

form

the

and

z

complete

library

A.

WDer

Conformational

analysis

of

backbone-independent

interactions

As

a

very

simple

model

for

steric

interactions

in

biopolymers

we

have

previously

compared

the

distribution

of

side-chain

rotarners

with

steric

interactions

that

are

well-documented

in

simple

hydrocarbons

such

as

butane

and

pentane

-LRB-

Dunbrack

Karplus

-RRB-

The

barrier

to

rotation

in

ethane

was

estimated

to

be

kcal/mol

as

early

as

the

's

-LRB-

Kemp

Pitzer

-RRB-

with

potential

energy

maxima

in

the

eclipsed

conformations

-LRB-

i.e.

with

a

dihedral

H-CC-H

-RRB-

and

minima

with

the

hydrogens

staggered

with

each

H-C-C-H

dihedral

equal

to

or

From

spectroscopic

data

and

the

experimental

thermodynamics

of

longer

hydrocarbons

it

was

also

clear

that

certain

combinations

of

neighboring

dihedrals

namely

g

g

and

g

g

in

longer

hydrocarbons

were

not

allowed

because

of

steric

hindrance

between

carbon

atom

i

and

i

of

the

chain

-LRB-

Pitzer

-RRB-

Recent

high-level

ab

initiocalculations

-LRB-

Wiberg

Murcko

-RRB-

and

experimental

measurements

-LRB-

Dung

Cornpton

Compton

et

al.

-RRB-

have

shown

that

the

single

gauche

interaction

in

butane

is

about

kcal/mol

higher

than

the

global

minimum

trans

conformation

-LRB-

-RRB-

The

abinitio

energy

of

two

consecutive

gauche

interactions

of

like

sign

in

pentane

has

an

energy

of

kcal/mol

above

the

syn-pentane

global

minimum

-LRB-

t

t

-RCB-

conformation

whereasthe

-LRB-

g

g

-RCB-

conformation

had

an

energy

of

kcal/mol

above

-LRB-

t

t

-RRB-

-LRB-

Wiberg

Murcko

-RRB-

Side-chain

y

and

heavy

atoms

-LRB-

of

the

ith

residue

-RRB-

can

interact

with

backbone

atoms

Ni

and

Ci

in

ways

that

are

dependent

on

the

values

of

xI

and

These

interactions

have

been

recognized

for

some

timeas

being

responsible

for

the

-LRB-

rl

rZ

-RRB-

rotamer

distribution

of

protein

side

chains

-LRB-

Chandrasekaran

Ramachandran

-RRB-

In

Figure

we

show

a

Newman

projection

of

-LRB-

rl

rz

-RCB-

rotamers

for

hydrocarbon

side

chains

-LRB-

Lys

Arg

Met

Glu

Gln

Leu

Val

Ile

-RRB-

In

Table

the

backbone-independent

gauche

and

syn-pentane

interactions

for

all

side-chain

types

are

listed

We

can

count

these

interactions

to

get

a

rough

estimate

of

steric

and

dihedral

strain

in

the

various

rotamer

combinations

Each

gauche

interaction

g

costs

about

kcal/mol

whereas

each

syn-pentane

interaction

costs

p

where

p

kcal/mol

We

also

include

the

CHAR

energies

in

Table

for

the

pentanoic

acid

-LRB-

Ape

side

chain

-LRB-

Ccr

-RRB-

-RRB-

ne

and

Leu

side

chains

These

energies

are

given

relative

to

the

lowest

energy

for

each

rl

rotamer

and

therefore

represent

the

interactions

responsible

for

the

parameters

of

the

conditional

backbone-independent

rotamer

library

described

above

The

rl

and

rotamers

of

Lys

Arg

Met

Glu

Gln

Leu

and

ne

are

exactly

analogous

to

the

g

t

and

g

rotamers

of

butane

and

pentane

because

for

these

side

chains

the

xI

and

dihedral

rotations

are

about

carbon-carbon

bonds

-LRB-

xl

Ccu-Cp

and

Cp-Cy

-RRB-

For

the

aromatic

side

chains

and

Asn

and

Asp

the

y

heavy

atom

has

an

hybridization

state

and

therefore

the

rotation

is

not

described

by

the

g

t

g

rotameric

states

The

energy

curves

of

the

peptide

fragment

of

pentanoic

as

acid

are

given

in

Figure

a

function

of

for

each

of

the

three

rl

rotamers

Four

of

the

nine

local

minima

have

syn-pentane

in

B.

Wbox

C.Wnon-per

Fig.

Weighting

functions

for

counting

backbone-dependent

rotamers

A

W

-LRB-

y

-LRB-

-RRB-

B

Wfr

-LRB-

r

-RRB-

C

W

-LRB-

e

-RRB-

for

-LRB-

I

h

The

functional

forms

are

given

in

the

text

x

angles

in

side

chainsdepend

on

a

number

of

variables

but

we

would

like

to

estimate

their

averages

conditional

upon

backbone

conformation

With

an

infinitely

large

database

we

would

simply

calculate

the

distribution

of

x

angles

-LRB-

j

I

j

-RCB-

as

a

funcThere

are

not

enough

data

in

the

datation

of

-LRB-

t

b

-RRB-

base

for

rare

and

even

not-so-rare

rotamers

to

determine

this

many

parameters

and

so

we

make

use

of

informative

prior

distributions

and

calculate

only

a

subset

of

the

possible

x

angle

parameters

The

prior

distributions

are

based

on

the

approximation

that

x

angle

averages

are

likely

to

depend

only

on

their

own

rotameric

state

and

the

rotameric

states

-LRB-

or

and

dihedrals

in

the

case

of

r

l

-RRB-

one

previous

in

the

chain

and

after

in

the

chain

As

an

example

we

one

show

the

one-dimensional

prior

distributions

for

xI

averages

as

a

function

of

in

Figure

for

all

several

side-chain

types

The

forms

of

these

functions

are

explained

easily

with

the

conformational

analysis

reviewed

in

the

next

section

We

combine

the

prior

distributions

for

the

xI

angles

as

shown

in

Figure

with

the

data

to

Downloaded

from

on

May

Published

by

Cold

Spring

Harbor

Laboratory

Press

R.L.

Dunbrack

J

and

EE

Cohen

I

I

c.

IO

P

Fig.

Data

calculated

with

the

nonperiodic

weighting

function

-LRB-

solid

bars

-RRB-

for

y

and

p

and

the

prior

density

function

from

W

-LRB-

lines

-RRB-

for

-LRB-

A

-RRB-

Arg

-LRB-

R

-RRB-

Arg

-LRB-

C

-RRB-

Phe

-LRB-

D

-RRB-

Phe

JI

-LRB-

E

-RRB-

Asp

-LRB-

F

-RRB-

Asp

-LRB-

G

-RRB-

Val

-LRB-

H

-RRB-

Val

JI

Regions

of

d

or

JI

where

interactions

between

backbone

atoms

and

Cy

are

likely

to

lower

the

populations

of

certain

rotamers

are

marked

with

arrows

Arrows

point

from

values

of

the

dihedral

connecting

the

backbone

atom

to

Cp

from

to

or

from

to

See

Table

-LRB-

Figure

conrinues

on

following

pages

-RRB-

teractions

when

N

and

CS

-LRB-

x

xr

-RRB-

are

connected

by

-LRB-

g

g

-RCB-

or

-LRB-

g

g

-RSB-

dihedrals

and

when

C

and

CS

-LRB-

xl

are

con

-RRB-

nected

by

-LRB-

g

g

-RSB-

or

-LRB-

g

g

-RSB-

dihedrals

These

four

are

evident

in

Figure

at

the

-LSB-

g

g

-LSB-

g

g

-LSB-

t

g

-RCB-

and

-LRB-

g

gf

-RSB-

positions

Three

of

the

remaining

five

rotamers

have

gauche

interactions

-LRB-

-LRB-

g

t

-RSB-

-LRB-

t

g

-RSB-

and

-LRB-

g

g

-RSB-

-RRB-

raising

the

energy

by

kcal/mol

over

the

global

minima

-LRB-

t

t

-RRB-

and

-LSB-

g

r

-RSB-

These

interactions

are

listed

in

Table

and

rotamer

pairs

with

synpentane

interactions

are

indicated

in

the

experimental

data

listed

in

Table

all

with

very

low

probabilities

and

with

skewed

x

angles

Ile

resembles

the

Ape

side

chain

except

that

the

rl

g

rotamer

is

lower

in

energy

than

the

other

two

rotamers

because

of

Downloaded

from

on

May

Published

by

Cold

Spring

Harbor

Laboratory

Press

Bavesian

statistics

of

protein

side

chains

P

gauche

interactions

with

backbone

Nj

and

C

-LRB-

equivalent

to

Val

t

rotamer

-RRB-

Leu

has

atoms

at

and

and

so

there

is

a

total

of

eight

syn-pentane

interactions

These

occur

for

all

three

r

l

g

rotamers

and

so

Leu

is

very

unlikely

to

be

ina

g

rotamer

The

conditional

probabilities

for

the

rl

g

rotamer

of

Leu

in

Table

shows

that

the

-LRB-

g

g

-RCB-

rotamer

has

much

lower

probability

than

the

other

two

g

gc

and

g

t

The

reason

is

evident

from

Table

because

the

-LRB-

g

g

-RRB-

combination

has

two

syn-pentane

interactions

with

the

backbone

whereas

the

other

two

r

l

g

A

rotamers

have

only

one

In

Figure

the

energies

of

Phe

rotamers

demonstrate

that

the

rl

rotamer

affects

the

position

of

the

local

xZminimum

Assuming

the

rotation

about

Ca-CP

has

minima

at

and

there

is

some

deviation

in

the

t

and

g

rl

rotamers

These

are

caused

by

interactions

of

the

atoms

with

backbone

N

and

C

when

rl

is

g

a

syn-pentane

interaction

when

is

below

pushes

the

minimum

to

yz

when

rl

is

r

a

syn-pentane

is

above

pushes

the

minimum

down

to

interaction

when

g

rotamers

are

always

higher

than

g

and

t

rotamers

xz

xZ

Downloaded

from

on

May

Published

by

Cold

Spring

Harbor

Laboratory

Press

R.L.

Dunbrack

J

and

EE

Cohen

i

P

by

more

than

kcal/mol

because

of

syn-pentane

interactions

between

and

and

N

and

C

We

note

that

typicalmolecularmechanicscalculationsfor

p

-LRB-

r

l

r

r

r

r

-LRB-

Nayeem

Scheraga

-RRB-

resemble

our

J

-RRB-

and

not

OlJkI

The

traditional

backbone-independent

rotamer

library

parameters

p

-LRB-

r

I

r

r

r

Jare

determined

both

by

the

relative

ener

-RRB-

gies

of

the

three

rotamers

in

each

backbone

conformation

and

by

the

populations

of

side

chains

in

different

backbone

conformations

-LRB-

i.e.

the

Ramachandran

map

distribution

-RRB-

They

can

be

calculated

by

integrating

the

product

of

the

conditional

backbone-independent

probabilities

and

the

backbone-dependent

rl

probabilities

over

the

whole

Ramachandran

map

E

-LRB-

Ova

-RRB-

p

-LRB-

r

l

i

i

i

r

i

-RRB-

Conformational

analvsis

of

backbone-conformation-dependent

interactions

Rotamer

populations

are

affected

not

only

by

backbone

N

and

C

of

residue

i

whose

positions

arc

independent

of

and

but

also

by

the

positions

of

other

backbone

atoms

especially

C

O

and

N

whose

positions

are

dependent

on

and

These

three

atoms

are

all

connected

to

y

heavy

atoms

by

two

hybridized

atoms

-LRB-

CP

and

C

a

-RRB-

and

one

sp

hybridized

atom

-LRB-

backbone

N

or

C

-RRB-

There

are

therefore

two

dihedral

degrees

of

freedom

separating

the

y

heavy

atoms

with

the

backbone

atoms

C

H

Oi

and

N

I

-LRB-

see

Fig.

for

the

Newman

projection

of

the

side-chain

with

the

dipeptide

backbone

added

-RRB-

We

include

H

and

its

hydrogen

bond

acceptor

because

there

are

apparent

shifts

in

rotamer

populations

due

to

this

interaction

Hence

there

is

the

possibility

of

syn-pentane

interactions

between

any

of

these

atoms

and

sidechain

y

atoms

when

the

connecting

dihedrals

occur

in

-LRB-

g

g

-RRB-

or

-LSB-

g

g

A

-RRB-

combinations

The

interactions

are

of

smaller

magnitude

because

bond

angles

at

the

sp

hybridized

atoms

are

instead

of

They

are

nevertheless

of

sufficient

magnitude

to

alter

the

rotamer

distributions

significantly

Thesyn-pentane

interactions

that

occur

between

y

heavy

atoms

and

backbone

atoms

dependent

on

and

I

are

listed

in

Table

along

with

the

probable

-RRB-

ranges

of

interaction

As

with

pentane

the

range

of

interaction

is

about

We

calculated

the

energies

of

the

rotamers

of

several

sidc

chains

to

investigate

how

well

the

CHARMM

potential

describes

the

experimental

distributions

of

the

backbone-dependent

rotamer

library

This

was

done

by

minimizing

the

energy

of

the

dipeptide

of

each

residue

type

-LRB-

N-acetyl-Xxx-N

methylamide

-RRB-

with

and

constrained

to

values

in

IO

increments

The

probability

surfaces

of

the

three

rotamers

of

butanoic

acid

-LRB-

A

h

-RRB-

and

valine

as

x

Downloaded

from

on

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Published

by

Cold

Spring

Harbor

Laboratory

Press

Bavesian

statistics

o

protein

side

chains

f

A

loo

m

B

L

Fig.

Bayesian

analysis

of

backbone-dependent

rotamer

populations

of

Arg

A

r

l

R

B

r

l

r

C

rl

x

Each

figure

shows

density

plots

of

the

data

calculated

with

the

nonperiodic

function

W

n

cr

data

-RRB-

-LRB-

the

Wpr

-LRB-

used

to

calculate

the

prior

distributions

-RRB-

the

prior

distributions

and

the

posterior

distributions

In

the

W

pPr

plots

regions

outside

the

contour

lines

represent

values

of

and

with

no

side

chains

in

the

database

a

function

of

and

b

are

shown

in

Figure

and

B

respectively

To

produce

these

figures

we

assumed

kT

When

rl

g

we

expect

a

syn-pentane

interaction

when

-LRB-

Cy

and

Ci

I

-RRB-

and

-LRB-

Cy

and

O

HNi

-RRB-

and

when

b

-LRB-

Cy

and

Ni

I

-RRB-

and

-LRB-

Cy

and

-RRB-

These

areshown

clearly

as

the

vertical

and

horizontal

darker

regions

in

the

first

panel

of

Figure

forthe

Abu

side-chain

These

are

also

evident

as

troughs

in

the

experimental

data

in

the

rl

I

panels

of

Figure

-LRB-

marked

by

arrows

-RRB-

These

interactions

also

appear

as

the

dark

regions

in

interthe

rl

panels

of

Figure

and

Figure

The

C

Y

C

action

is

the

largest

approximately

kcal/mol

whereas

the

C

y

Ni

I

and

C

y

O

jinteractions

are

and

kcal/mol

respectively

In

the

second

panel

of

Figure

the

interactions

o

f

t

rotamer

Cy

with

Ni

I

when

b

and

Cy

with

Oi

when

b

are

also

evident

These

probability

minima

are

also

evident

experimentally

in

the

rl

panels

of

Figures

and

Again

these

interactions

are

marked

by

arrows

in

Figure

The

third

panel

of

Figure

confirms

the

C

y

C

i

I

interaction

when

c

and

near

for

the

g

rl

rotamer

as

shown

in

the

data

in

Figures

and

It

is

worth

noting

the

weak

b

dependence

in

the

g

panels

reflecting

changes

in

the

energies

of

the

other

two

rotamers

as

a

function

of

b.It

is

clear

from

the

three

maps

that

the

x

conformation

of

Abu

is

the

most

favored

in

much

of

the

heavily

occupied

portions

of

the

Ramachandran

map

-LRB-

Le.

c

b

and

b

-RRB-

rl

t

has

steric

interactions

with

backbone

atoms

in

extended

backbone

conformations

and

Downloaded

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by

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Harbor

Laboratory

Press

R.L.

Dunhrack

J

and

EE

Cohen

-LRB-

AGfi

i

v

kcal/mol

-RRB-

It

would

seem

that

a

protein

would

not

be

able

to

afford

more

than

one

or

two

such

side-chain

conformations

and

it

would

therefore

be

useful

to

scrutinize

the

electron

density

carefully

in

such

cases

to

determine

if

there

are

interactions

that

explain

the

high-energy

conformation

or

whether

a

more

plausible

conformation

can

be

found

-LRB-

Schrauber

et

al.

-RRB-

These

results

can

be

considered

minimum

values

for

any

sidechain

prediction

method

for

the

self-backbone

prediction

test

because

they

were

obtained

with

no

computational

effort

beyond

looking

up

the

best

rotamer

in

the

database

We

have

recently

used

the

backbone-dependent

rotamer

library

described

in

this

paper

in

a

prediction

program

that

removes

steric

clashes

after

the

backbonedependent

rotamer

library

prediction

-LRB-

Bower

et

al.

-RRB-

Removing

steric

clashes

in

a

set

of

self-backbone

side-chain

prediction

tests

raises

the

prediction

rate

from

to

We

compared

X-ray

structures

of

identical

proteins

in

different

crystal

space

groups

and

found

that

only

of

side

chains

retain

their

rl

rotamers

in

different

crystal

structures

It

is

likely

that

a

large

fraction

of

the

remaining

are

in

two

different

rl

rotamers

one

of

which

is

the

lowest

energy

rotamer

for

the

local

backbone

conformation

Discussion

Understanding

the

factors

that

determine

protein

side-chain

conformation

and

its

dynamics

is

important

in

a

number

of

areas

First

the

fact

that

the

backbone

has

such

a

strong

effect

on

side-chain

conformation

distributions

is

likely

to

be

exploited

by

proteins

as

they

fold

This

effect

may

also

operate

as

proteins

evolve

if

mutations

of

residues

with

unfavorable

steric

interactions

to

residues

without

these

interactions

increase

the

stability

of

the

folded

protein

This

topic

has

been

studied

extensively

using

site-directed

mutagenesis

to

probe

the

structural

and

thermodynamic

implications

of

a

side

chain

's

interaction

with

its

environment

Mutations

that

increase

or

decrease

local

backbone-side-chain

steric

interactions

might

be

expected

to

have

some

effect

on

the

stability

of

the

folded

structure

because

these

interactions

are

presumably

absent

in

the

unfolded

state

Second

homology

modeling

of

proteins

for

the

purposes

of

drug

design

depends

mostly

upon

an

accurate

representation

of

the

sidechain

conformations

in

the

target

structure

It

has

proved

very

difficult

to

model

the

insertion

and

deletion

of

loops

in

proteins

but

in

cases

where

these

insertions

are

far

from

the

binding

site

of

an

enzyme

substrate

or

allosteric

effector

it

is

only

the

binding

site

side-chain

conformations

that

must

be

constructed

carefully

Because

proteins

favor

their

dipeptide

low-energy

rotamers

in

their

folded

states

the

backbone-dependent

rotamer

library

is

especially

helpful

in

predicting

side-chain

conformations

in

homology

modeling

situations

where

no

information

is

present

in

the

template

structure

-LRB-

e.g.

Ala

Val

Pro

Phe

-RRB-

It

is

notable

thatmany

current

side-chain

placement

methods

use

rotamer

libraries

dependent

on

the

local

structural

environment

of

side

chains

-LRB-

Moult

James

Levitt

Bower

et

al.

-RRB-

including

those

popular

molecular

modeling

packages

e.g.

used

in

several

QUANTA

-LRB-

Dunbrack

Karplus

-RRB-

SYBYL

-LRB-

Schrauber

et

al.

-RRB-

and

WHAT-IF

-LRB-

Chinea

et

al.

-RRB-

We

have

set

up

a

website

of

the

conformational

analysis

of

protein

side

chains

-LRB-

Dunbrack

-RRB-

as

described

in

this

paper

The

website

includes

both

backbone-independent

and

backbone

Fig.

Average

backbone-dependent

posterior

parameters

for

Asp

Phe

and

Val

The

averages

are

given

by

the

expectation

values

of

Equation

near

b

O

whereas

rl

has

steric

conflicts

in

P-sheets

as

well

as

a-helices

In

Figure

the

CHARMM

probability

and

energy

surfaces

for

Val

are

shown

The

alternation

between

t

and

rotamers

with

variation

in

b

isdue

to

interactions

with

each

y

carbon

with

backbone

N

I

and

which

result

in

the

troughs

at

b

and

for

the

second

panel

-LRB-

t

-RRB-

and

at

b

and

in

the

third

panel

-LRB-

g

-RRB-

Both

of

these

rotamers

have

steric

interactions

with

Ci

I

at

J

and

so

the

g

rotamer

in

the

first

panel

has

high

density

in

this

region

These

figures

can

be

compared

with

the

experimental

probabilities

from

the

Bayesian

analysis

for

Valin

Figure

and

H

and

Figure

Finally

inTable

we

show

the

prediction

rates

for

the

backbonedependent

rotamer

library

The

data

in

this

table

were

obtained

by

predicting

the

side-chain

conformation

for

each

side

chain

in

the

database

based

on

the

identity

of

the

rI

rotamer

with

the

largest

probability

in

the

posterior

distribution

given

the

values

of

J

and

b

for

that

side

chain

No

optimization

or

removal

of

steric

clashes

was

performed

The

library

is

able

to

predict

of

xi

's

of

side

chains

that

have

structures

within

of

one

of

the

canonical

rotamers

-LRB-

-RRB-

From

the

steric

analysis

and

the

CHARMM

calculations

it

seems

that

most

of

these

are

sterically

the

lowest

energy

minima

-LRB-

Le.

ignoring

electrostatic

interactions

in

polar

sidechains

-RRB-

given

the

local

backbone

dihedral

angles

J

and

Another

of

side

chains

adopt

the

next

most

likely

local

minimum

energy

structure

and

the

final

the

highest

energy

least

likely

local

minimum

Close

to

of

all

side

chains

are

not

within

of

one

of

the

g

g

and

t

rotamer

conformations

It

is

likely

that

most

of

these

are

either

an

average

of

two

occupied

rotamers

or

an

error

of

interpretation

of

electron

density

given

the

high

energy

price

paid

-LRB-

kcal/mol

-RRB-

for

these

strained

conformations

and

that

proteins

are

only

marginally

stable

e.

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Laboratory

Press

Bayesian

statistics

of

protein

side

chains

b

-LRB-

Ln

d

-LRB-

v

J

l

cu

x

l

a

b

l

z

a

-LRB-

v

b

-LRB-

v

I

I

I

w

Fig.

Average

y

angle

dependent

prior

distributions

for

aromatic

side

chains

The

values

given

for

each

angle

are

deviations

from

the

canonical

values

of

and

viz.

yI

yI

and

yI

for

the

rl

rotamers

Arrows

indicate

ranges

of

syn-pentane

interactions

between

side-chain

Cy

and

backbone

atoms

dependent

analysis

as

well

as

raw

data

counts

and

the

Bayesian

posterior

distributions

described

here

We

have

shown

graphical

representations

of

our

results

for

several

side

chains

in

this

paper

On

the

website

we

make

available

the

graphical

representations

for

all

side-chain

types

As

the

PDB

continues

to

grow

we

will

update

the

results

presented

here

periodically

and

place

them

on

the

website

Although

many

of

the

steric

effects

in

backbone-independent

and

dependent

side-chain

conformations

described

in

this

paper

have

been

studied

previously

in

protein

structures

-LRB-

Chandrasekaran

Ramachandran

Cody

et

al.

Janin

et

al.

Bhat

et

al.

Benedetti

et

al.

James

Sielecki

Sutcliffe

McGregor

et

al.

Ponder

Richards

et

Tuffery

et

al.

-RRB-

and

by

energy

calculations

-LRB-

Pon

Downloaded

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Cold

Spring

Harbor

Laboratory

Press

R.L.

Dunbrack

J

and

FE

Cohen

A

B

HCi

Fig.

Newman

projections

for

conformational

analysis

of

protein

side

chains

A

Backbone-independent

interactions

B

Backbonedependent

interactions

Dotted

lines

mark

possible

syn-pentane

interactions

nuswamy

Sasisekharan

Sasisekharan

Ponnuswamy

Lewis

et

al.

Janin

et

al.

Zimmerman

Scheraga

Gelin

Karplus

Benedetti

et

al.

-RRB-

we

believe

the

present

analysis

is

more

thorough

and

consistent

in

its

approach

In

particular

the

conformational

analysis

of

gauche

and

syn-pentane

interactions

has

provided

a

simple

organizing

principle

for

explaining

and

predicting

the

effects

of

backbone

conformation

on

side-chain

conformation

that

previously

has

not

been

used

extensively

on

peptides

We

hope

the

library

will

be

useful

in

a

number

of

applications

as

well

as

furthering

our

understanding

of

the

determinants

of

protein

conformation

and

the

process

of

protein

folding

j

k

Yijkl

p

-LRB-

r

g

j

-RRB-

Pkl

Yijkl

i

k

Materials

and

methods

Experimental

data

To

update

the

backbone-dependent

rotamer

library

we

obtained

a

list

of

protein

chains

with

resolution

or

better

and

less

than

sequence

identity

with

other

chains

in

the

list

from

the

OBSTRUCT

server

of

Heringa

et

al.

-LRB-

-RRB-

The

present

library

is

based

on

chains

which

is

four

times

larger

than

the

original

library

The

list

of

proteins

used

is

available

on

the

BackboneDependent

Rotamer

Library

Website

-LRB-

Dunbrack

-RRB-

and

Rotamers

r

l

r

r

g

were

defined

according

to

the

limits

on

y

angles

listed

in

Table

for

each

amino

acid

type

Residues

with

missing

backbone

atoms

necessary

to

determine

J

and

-LRB-

i.e.

the

first

and

last

residue

of

a

chain

-RRB-

or

with

missing

side-chain

atoms

were

discarded

As

in

the

previous

work

we

assumed

that

in

crystal

structures

the

orientations

of

the

His

and

Asn

side

chains

and

the

orientation

of

Gln

are

not

uniquely

determined

under

a

rotation

of

For

the

raw

backbone-independent

library

we

used

the

limits

in

Table

to

count

the

number

of

side

chains

with

rl

i

j

k

l

for

all

side-chain

types

These

numbers

Y

i

j

k

l

can

be

used

to

calculate

various

raw

conditional

probabilities

which

will

be

used

in

the

Bayesian

analysis

To

be

consistent

with

our

previous

work

-LRB-

Dunbrack

Karplus

-RRB-

we

first

calculated

a

raw

count

of

each

rl

in

by

J

centered

apart

-LRB-

bins

-RRB-

in

J

and

Each

sidechain

is

counted

four

times

in

such

a

procedure

Mathematically

we

can

define

a

function

Wboxfor

a

bin

centered

at

-LRB-

J

o

b

-RRB-

that

is

applied

to

all

side

chains

with

rotamer

rl

i

such

that

I

where

Adm

#Jm

#Ja

and

AGm

@b

Wbox

if

both

A

#Jmand

A@mare

less

than

W

box

otherwise

The

function

w

box

isshowninFigure

for

-LCB-

J

a

b

-RRB-

-LCB-

The

jry

Tzb

corresponding

probabilities

are

denoted

y

z/z

To

provide

estimates

for

the

underpopulated

and

unpopulated

regions

of

the

map

we

used

a

Gaussian

weighted

periodic

function

to

count

rotamers

Side

chains

were

counted

for

each

rl

rotamer

bins

type

with

the

following

weight

function

in

J

centered

apart

-LRB-

-RRB-

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Press

Bayesian

statistics

of

protein

side

chains

Table

Conformational

analysis

of

backbone-independent

interactions

Single

Cy

single

-LRB-

Met

Glu

Gln

Arg

Lys

-RRB-

Met

Glu

Gln

Arg

Lys

rotamers

x

Ca-CP-Cy-CS

g

Eb

AE

CHARMM

AE

Argd

Met

Gln

Glu

LY

s

Ca-CP-Cyl-CG

N-Ca-CP-Cyl-CS

C-Ca-CP-Cy

E

AE

XI

XI

XI

x

XI

x

Xz

P

P

P

g

CHARMM

A

E

Ile

E

AE

CHARMM

A

E

Leu

g

g

P

g

p

-LRB-

-RRB-

a

p

denotes

a

syn-pentane

interaction

between

the

atom

and

of

the

listed

in

the

first

column

kcal/mol

based

on

pentane

ab

initio

calculations

-LRB-

Wiberg

Murcko

bAs

a

rough

estimate

one

gauche

-LRB-

g

-RRB-

interaction

is

kcal/mol

p

-RRB-

CHAR

energies

are

given

relative

to

the

lowest

rotamer

for

each

r

l

rotamer

dExperimental

data

for

the

conditional

probabilities

p

-LRB-

-RRB-

Y$

sidechains

m

with

rl

t

x

exp

-LRB-

kl

-LRB-

A

m

-RRB-

k

-LRB-

AtJm

-RRB-

-RRB-

-LRB-

-RRB-

xexp

-LRB-

-LRB-

T

z

-LRB-

Fr

-RRB-

E

N-Ca-CP-Cy

C-Ca-CP-Cy

XI

g

g

g

g

t

g

g

t

g

t

t

g

g

t

g

g

t

g

t

g

t

g

g

t

P

P

P

t

gt

t

g

g

t

t

g

f

g

t

P

P

P

P

g

P

P

g

P

P

P

P

P

P

P

ppp

g

g

P

P

g

p

g

P

g

P

k

is

a

constant

that

determines

the

width

of

the

function

about

-LRB-

C#

J

-RRB-

determines

the

ratio

of

the

peak

heights

at

-LCB-

t

hb

-RCB-

and

-LRB-

@b

-RCB-

-LRB-

t

-RRB-

b

-RRB-

and

fb

-RCB-

We

used

values

of

kl

and

radians

This

function

is

also

plotted

in

Figure

for

-LRB-

f

ba

b

-RCB-

-LRB-

O

-RCB-

The

corre

Downloaded

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Laboratory

Press

R.L.

Dunbrack

JK

and

EE

Cohen

A

B

I

I

I

I

I

I

CTI

v

E

h

E

Y

w

Y

w

Fig.

CHARMM

calculations

of

backbone-conformation

independent

backbone-side-chain

interactions

A

Ape

B

Phe

sponding

probabilities

p

were

calculated

by

normalizing

Equation

for

each

-LRB-

bo

b

pair

y

Ziry

We

used

a

third

function

W

n

n-ppr

to

count

rotamers

for

the

data

used

in

the

Bayesian

analysis

This

function

is

a

product

of

W

p

e

r

It

and

Wbux

counts

side

chains

with

an

approximately

Gaussian

weight

in

the

neighborhood

of

each

-LRB-

b

point

But

because

of

the

box

function

Wb

x

side

chains

with

A

-LRB-

b

or

A

greater

than

are

not

counted

by

W

n

o

n

p

r

r

-LRB-

see

Fig.

-RRB-

Bayesian

statistical

analysis

Rotamer

statistics

We

derive

a

library

used

for

predictions

of

side-chain

rotmers

from

the

PDB

by

a

Bayesian

statistics

analysis

-LRB-

Gelman

et

d.

-RRB-

In

this

section

we

describe

Bayesian

statistical

analysis

briefly

further

details

for

the

rotarner

library

are

described

below

We

follow

the

notation

of

Gelman

et

al.

-LRB-

-RRB-

We

are

considering

Table

Conformational

analysis

of

backbone-dependent

interactions

Atoms

Connecting

dihedrals

Interaction

f

O

g

Interaction

g

g

g

N

N

Ca

Ca

c

c

N

N

ca

Ca

Ca

Ca

Cp

Cp

cp

cp

Cp

Cp

Cy

Cy

cy

cy

All

Cy

Cys

Sy

Ser

Oy

Val

Cyl

Ile

Cyl

Thr

Oyl

XI

rl

g

XI

rl

g

rI

t

X

I

IC

rl

t

X

I

II

Val

X

XI

I

CL

r

gI

rl

g

rl

g

rl

g

rl

t

c

c

N

N

ca

Ca

Ca

Ca

cp

cp

Cp

Cp

I

-RRB-

rl

grl

g

rl

g

rl

g

rl

t

IC

rl

t

rl

g

rl

g

rl

g

c

c

ca

ca

cp

cp

Ile

and

Thr

I

rl

t

rl

grl

g

r

g

I

rl

t

rl

f

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Bayesian

statistics

of

protein

side

chains

B

Fig.

CHARMM

calculations

of

hackhone-conformation

dependent

hackhone-sidc-chain

interactions

A

Ahu

prohahilitics

R

Val

prohahilities

Prohahilities

werc

calculated

from

Equation

assuming

kT

a

population

of

side

chains

in

different

states

or

rotamers

We

refer

to

the

probabilities

of

the

different

rotamers

as

so

for

the

three

rl

rotamers

we

have

and

These

are

superpopulation

parameters

that

is

the

probabilities

of

the

three

different

rotamers

in

a

hypothetical

infinitely

large

Protein

Data

Bank

The

observed

data

yI

and

are

the

numbers

of

rotamers

observed

in

the

finite

experimental

data

of

the

PDB

Unless

the

total

of

yI

v

N

is

large

-LRB-

-RRB-

the

observed

populations

y

J

N

may

differ

somewhat

from

the

Bayesian

analysis

provides

a

method

for

calculating

a

probability

distribution

of

thc

parameters

based

on

the

ohserved

data

y

where

and

y

are

vectors

with

m

components

-LRB-

m

in

this

example

-RRB-

We

start

with

the

joint

probnbilir

distribution

for

and

y

written

with

Bayes

rule

as

the

product

of

a

prior

distribution

p

-LRB-

-RRB-

and

the

likelihood

or

santplin

distribution

p

-LRB-

V

-RRB-

p

-LRB-

v

-RRB-

P

-LRB-

o

-RRB-

P

-LRB-

vlo

-RRB-

Because

y

is

fixed

we

can

write

the

posterior

distribution

p

-LRB-

-RRB-

as

proportional

to

the

unnormalized

posterior

densie

-LRB-

-RRB-

So

if

we

have

some

guess

for

the

prior

distribution

p

-LRB-

-RRB-

and

a

model

for

the

likelihood

function

p

-LRB-

-RRB-

for

the

probability

of

the

data

given

values

for

the

parameters

wecan

derive

a

posterior

probability

distribution

for

the

parameters

p

-LRB-

Ol

v

-RRB-

There

is

a

distinction

in

Bayesian

statistics

between

hformative

and

noninfonnative

prior

distributions

Noninformative

prior

distributions

are

usually

flat

functions

of

the

parameters

Informativc

prior

distributions

are

defined

by

probabilities

that

are

higher

in

certain

ranges

of

the

parameters

where

we

expect

the

value

of

to

lie

and

probabilities

that

are

lower

in

less

likely

ranges

For

the

rotamer

problem

we

use

a

multinomial

model

such

that

thc

likelihood

takes

the

form

The

prior

distribution

contains

any

information

we

may

have

about

the

expected

value

of

in

the

form

of

a

probability

density

over

the

acceptable

values

of

We

are

looking

for

an

expression

forp

-LRB-

BI

I

-RRB-

which

is

the

probability

distribution

of

the

parameters

of

interest

B

conditioned

on

the

observed

experimental

data

y.

Using

Bayes

rule

again

such

that

p

-LRB-

y

-RRB-

p

-LRB-

v

-RRB-

p

-LRB-

eIy

-RRB-

we

find

I

y

the

sum

of

the

counts

of

the

different

rotamers

where

v

and

C

B

We

use

a

conjugate

prior

distribution

-LSB-

a

prior

distribution

that

has

a

similar

mathematical

form

to

the

sampling

distribution

-LRB-

Equation

I

-RRB-

-RSB-

called

the

Dirichlet

distribution

x

Downloaded

from

on

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by

Cold

Spring

Harbor

Laboratory

Press

Table

P-values

and

prediction

rates

for

backbone-dependent

mtamers

prediction

prediction

prediction

N

binsb

p-value

R.L.

Dunbrack

JK

and

RE

Cohen

fore

simple

expressions

for

the

expectation

values

for

the

their

variances

covariances

and

modes

xi

y

Yo

m

Residue

N

Asn

ASP

CYS

Gln

Glu

His

Ile

Leu

L

s

Y

Met

Phe

Ser

Thr

Trp

Val

All

o

155 222 229 113

146 162 149 139

166 181 104 167

xi

Y

Yo

xo

yo

m

l

where

m

isthe

m

-RRB-

number

of

rotamers

under

consideration

-LRB-

i

aPrediction

rates

determined

by

th

ep

-LRB-

r

l

for

all

side

chains

in

the

database

First

prediction

is

made

based

on

the

highest

probability

rl

rotamer

from

the

appropriate

c

bin

second

prediction

is

the

next

highest

probability

third

prediction

is

based

on

the

lowest

probability

rl

rotamer

bNumber

of

bins

with

v

P

p

r

I

O

used

to

calculate

the

p-values

in

the

last

column

where

x.

C

x

i

Note

that

for

an

integer

T

-LRB-

n

-RRB-

n

The

values

of

the

hyperparameters

xi

that

define

the

prior

distribution

can

be

thought

of

as

estimated

counts

for

the

rotamers

in

some

sample

of

side

chains

of

size

xo

These

estimates

can

come

from

any

source

including

previous

data

or

some

pooling

of

the

present

data

The

total

number

of

prior

counts

xo

can

be

scaled

to

any

value

to

alter

the

dependence

of

the

posterior

distribution

on

the

prior

distribution

The

larger

x

is

the

more

precise

the

prior

dis

tribution

is

and

the

closer

the

posterior

density

is

to

values

near

x

xo

The

posterior

distribution

that

results

from

Equations

and

is

also

Dirichlet

with

parameters

x

y

i.e.

Bayesian

simulation

of

the

posterior

distribution

Bayesian

inference

is

frequently

performed

by

means

of

simulation

especially

when

prior

distributions

and

sampling

distributions

are

complicated

and

do

not

produce

an

analytical

posterior

distribution

We

use

such

a

technique

as

a

means

for

finding

the

best

prior

distribution

To

check

our

models

against

the

experimental

data

it

is

useful

to

make

predictions

of

the

data

from

the

posterior

distribution

of

the

parameters

We

refer

to

these

as

yreP

for

L

for

L

draws

from

the

posterior

distribution

To

perform

simulations

we

first

draw

values

for

Or

from

a

Dirichlet

distribution

with

parameters

x

y

l

and

then

draw

yr

P

from

the

multinomial

likelihood

function

Equation

With

the

y

r

p

P

in

hand

we

can

evaluate

our

models

with

the

calculation

of

p-values

within

the

Bayesian

statistics

framework

Because

Bayesian

inference

is

performed

over

the

entire

posterior

distribution

of

the

parameters

rather

than

just

a

point

value

as

in

classical

statistics

we

calculate

the

proportion

of

simulation

draws

where

the

replicated

data

are

more

extreme

than

the

experimental

data

L

p

value

A

L

I

-LRB-

T

-LRB-

yrP

P

-RRB-

T

-LRB-

y

f

F

-RRB-

-RCB-

-RRB-

-LRB-

I

where

the

indicator

function

I

if

the

expression

inside

is

true

and

otherwise

T

is

the

test

statistic

which

is

a

function

of

both

y

and

We

use

a

y

discrepancy

which

is

similar

to

the

classical

y

goodness-of-fit

measure

where

the

sum

is

over

the

different

rotamer

types

and

y

is

either

from

the

experimental

data

or

the

replicated

data

The

use

of

the

conjugate

prior

distribution

results

in

the

analytical

form

of

the

posterior

distribution

-LRB-

Equation

-RRB-

and

also

thereWe

use

the

notation

from

the

following

table

for

the

counts

and

proportions

for

the

prior

data

and

posterior

distributions

Prior

Data

Posterior

Rotamer

libraries

derived

from

Bayesian

statistical

analysis

Backbone-independent

rotamer

library

For

the

backbone-independent

rotamer

library

we

use

informative

prior

distributions

of

the

form

c

r

k

J

Downloaded

from

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by

Cold

Spring

Harbor

Laboratory

Press

Bayesian

statistics

of

protein

side

chains

K

is

a

scale

factorthat

affects

the

influence

of

the

prior

distribution

on

the

posterior

distribution

by

setting

the

value

of

in

Equation

proportional

to

the

number

of

side

chains

yo

in

the

experimental

data

with

rl

i.

For

larger

values

of

K

the

prior

distribution

exerts

a

stronger

pull

on

the

experimental

data

Because

some

of

the

rotamer

types

are

not

seen

at

all

in

the

limited

data

set

we

would

like

some

estimate

of

their

probability

beyond

the

crudenoninformative

distribution

result

Higher

values

of

K

are

therefore

desirable

We

do

not

know

for

sure

how

good

our

prior

distribution

is

and

the

resulting

posterior

distribution

-LRB-

which

is

the

goal

-RRB-

is

dependent

on

it

So

we

test

the

posterior

distributions

using

the

Bayesian

version

of

the

x

test

by

simulating

draws

@rep

from

the

posterior

distributions

from

a

range

of

values

for

K

and

y

from

the

likelihood

functions

The

resulting

p-values

indicate

what

fraction

of

the

sample

draws

yielded

a

value

of

the

test

x

function

that

were

larger

than

the

value

for

the

actual

data

sample

Values

near

indicate

that

the

experimental

data

are

exactly

in

the

middle

of

the

posterior

distribution

Values

below

and

above

indicate

that

the

data

do

not

appear

to

come

from

the

posterior

distribution

We

divided

the

data

in

half

and

derived

backbone-independent

rotamer

libraries

from

each

half

database

as

well

as

the

full

database

For

values

of

K

when

the

data

and

posterior

distribution

come

from

the

same

data

set

the

p-values

stay

within

the

range

Using

as

large

a

value

of

K

as

possible

allows

us

to

emphasize

the

prior

distribution

in

cases

where

N

is

small

When

either

half

database

was

used

for

the

posterior

distribution

and

the

full

database

was

used

as

the

data

the

range

of

p-values

was

lower

but

they

increased

to

more

median

levels

as

K

was

increased

This

indicates

that

for

situations

with

less

experimental

data

-LRB-

the

half

databases

-RRB-

the

posterior

distribution

should

rely

heavily

on

an

informative

prior

distribution

We

found

a

value

of

K

to

be

optimal

This

is

a

useful

guideline

for

the

more

complicated

case

of

the

backbone-dependent

distributions

where

there

is

much

less

data

per

parameter

than

in

the

backbone-independent

case

Backbone-dependent

rotamer

library

We

would

like

to

determine

a

backbone-dependent

library

that

is

continuousover

the

full

Ramachandran

map

and

that

provides

good

estimates

for

regions

that

are

only

weakly

populated

or

even

unpopulated

in

the

PDB

We

can

check

the

model

in

various

ways

to

ensure

that

the

populated

regions

are

represented

accurately

In

our

prior

distribution

we

postulate

that

The

two-dimensional

prior

distribution

is

a

Dirichlet

function

at

and

I

b

-RRB-

each

value

of

where

the

and

dependent

probabilities

are

determined

from

the

SVD

equations

-LRB-

Equations

and

-RRB-

with

the

dependent

data

obtained

with

weighting

function

Wper.We

use

a

value

for

Xolab

in

a

limited

range

such

that

KYOlab

loo

KYOlab

where

K

is

whatever

proportionality

constant

we

choose

The

posteriordistribution

then

is

alsoDirichlet

asinthe

backbone-independent

case

with

Y

lab

obtained

from

the

nonperiodic

function

Wnon-per

The

use

of

W

n

o

n

p

e

r

here

minimizes

the

effect

of

the

periodic

function

in

populated

regions

of

the

map

To

test

how

well

the

posterior

distributions

represent

the

data

in

the

occupied

regions

of

the

Ramachandran

map

we

calculated

the

p-values

for

the

distributions

for

all

side

chains

with

the

results

listed

in

Table

The

full

rotamer

library

Given

the

strong

dependence

of

the

xIrotamer

probabilities

on

the

backbone

dihedrals

and

and

the

lack

of

dependence

of

on

and

-LRB-

see

Results

-RRB-

we

present

the

backbone-dependent

rotamer

library

as

the

conditional

probabilities

of

X

rotamers

given

The

full

the

values

of

the

backbone

dihedrals

and

p

-LRB-

r

l

-RRB-

rotamer

library

is

formed

by

combining

-LRB-

and

normalizing

-RRB-

the

conditional

backbone-independent

probabilities

for

with

P

-LRB-

I

-RRB-

p

-LRB-

r

l

i

j

r

k

ba

-RRB-

In

practice

a

robust

estimate

of

prior

distributionparameters

p

-LRB-

r

I

-RRB-

ailoandp

-LRB-

rl

i

-LRB-

b

-RRB-

-LRB-

Yjlb

can

be

obtained

from

a

loglinear

model

of

the

data

probabilities

p

-LRB-

rl

rclb

-RRB-

P$

This

is

an

overdetermined

problem

because

there

are

X

values

on

theleft-hand

side

to

be

expressed

as

a

linear

combination

of

values

on

the

right-hand

side

The

matrix

equation

can

be

solved

by

singular

value

decomposition

-LRB-

SVD

-RRB-

and

is

guaranteed

to

be

the

best

solution

of

the

overdetermined

problem

in

the

leastsquares

sense

-LRB-

Press

et

al.

-RRB-

from

a

Bayesian

analysis

using

a

normal

model

for

both

the

priorand

posterior

distributions

We

are

looking

for

average

x

angles

as

a

function

of

and

for

all

rotamer

combinations

e.g.

for

Lys

values

for

X

I

z

z

j

b

There

is

not

sufficient

information

in

the

PDB

r

to

determine

this

many

parameters

so

we

determine

only

a

subset

of

these

values

assuming

for

instance

that

X

and

j

are

independent

of

and

In

Table

we

list

the

parameters

that

were

used

in

theprior

and

posterior

distributions

for

the

x

angle

averages

Generally

we

expect

that

each

x

angle

's

distribution

is

affected

only

by

its

rotameric

state

and

the

dihedral

or

rotameric

state

on

either

side

This

is

reflected

in

the

choice

of

prior

distribution

dependencies

Some

non-neighboring

dependencies

are

included

in

the

posterior

distributions

in

case

they

are

of

some

relevance

in

the

data

analysis

x

Angle

averages

x

angle

averages

were

determined

Downloaded

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by

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Spring

Harbor

Laboratory

Press

Table

Dependence

of

and

rotamer

types

Residues

R.L.

Dunbrack

Jr.

and

EE

Cohen

x

angles

on

backbone

dihedrals

Posterior

distribution

distribution

Prior

The

assumption

that

dihedral

averages

depend

only

on

the

neighboring

rotameric

states

can

be

tested

by

calculating

test

statistics

as

described

above

It

should

be

noted

that

ujara

the

variance

in

is

the

x

angle

experimental

data

whereas

a

is

the

variance

in

the

mean

of

the

x

angle

i.e.

an

expression

of

our

uncertainty

in

the

mean

value

not

the

variance

in

the

data

from

an

infinitely

large

PDB

Molecular

mechanics

calculations

We

have

used

the

program

C

H

A

R

-LRB-

Brooks

et

al.

-RRB-

to

calculate

the

energies

of

side-chain

rotamers

in

both

the

backboneindependent

and

backbone-dependent

contexts

For

all

of

these

calculations

we

have

used

the

C

H

A

R

potential

energy

function

which

includes

all

atoms

-LRB-

polar

and

nonpolar

hydrogens

-RRB-

and

which

has

been

optimized

to

represent

a

variety

of

intramolecular

and

intermolecular

interactions

in

proteins

The

backbone-independent

energies

were

calculated

by

considering

the

atoms

in

a

single

residue

fragment

consisting

of

only

N

Ccr

Hcr

C

and

the

residue

side-chain

Backbone-dependent

rotamer

preferences

were

calculated

with

unconstrained

x

I

-LRB-

and

-RRB-

dihedrals

starting

from

near

the

likely

minima

-LRB-

-RRB-

by

fixing

the

and

dihedrals

of

the

N-acetyl

N

methylamide

of

each

amino

acid

with

force

constants

of

kcal/mol

and

minimizing

with

steps

of

the

conjugate

gradient

minimizer

We

placed

an

oxygen

atom

bonded

to

NH

of

each

dipeptide

in

the

position

of

a

likely

hydrogen

bond

acceptor

The

oxygen

was

fixed

at

a

distance

of

from

the

backbone

N

and

collinear

with

the

NH

bond

The

dielectric

constant

E

was

set

to

In

most

cases

the

minimized

were

less

than

from

the

starting

conformations

In

others

no

local

minimum

value

was

over

from

the

original

was

found

and

the

final

value

If

the

minimized

dihedral

angle

value

was

more

than

away

from

the

initial

value

a

force

constant

of

kcal/mol

was

applied

to

the

x

dihedrals

at

their

initial

values

and

the

minimization

was

repeated

This

ensures

that

the

rotamer

energies

-LRB-

after

excluding

the

constraint

energies

-RRB-

and

calculated

probabilities

correspond

to

the

correct

rotamers

To

remove

the

influence

of

the

backbone-backbone

steric

interactions

and

to

plot

the

database

energies

and

CHARMM

energies

-LRB-

b

-RRB-

in

a

similar

way

we

invert

the

Boltzmann

calculation

Pro

For

each

term

in

the

prior

distributions

we

use

a

normal

distribution

by

calculating

a

mean

and

variance

of

the

x

angle

from

side

chains

in

the

database

with

the

given

values

of

rl

etc.

If

we

have

two

different

prior

distributions

for

a

parameter

p

l

-LRB-

-RRB-

and

-LRB-

-RRB-

we

can

estimate

the

joint

prior

distribution

as

their

product

p

-LRB-

-RRB-

pl

-LRB-

-RRB-

-LRB-

-RRB-

So

for

instance

in

combining

the

and

dependencies

for

Cys

Ser

Val

and

Thr

x

where

the

values

of

p

and

u

come

from

the

data

distribution

By

completing

the

square

we

find

that

We

calculate

prior

distributions

based

on

the

expressions

in

Table

and

Equation

to

yield

a

normal

distribution

of

mean

Xprior

and

variance

u

posterior

distribution

is

then

calcuThe

lated

by

adding

in

the

data

with

all

the

correct

rotameric

states

as

and

defined

in

Table

yielding

values

for

u

The

posterior

distribution

is

normal

p

-LRB-

elY

-RRB-

-LRB-

I

X

p

o

s

r

L

-RRB-

-LRB-

-RRB-

The

value

of

kT

was

set

to

Graphical

analysis

was

performed

with

the

Trellis

graphics

module

-LRB-

Becker

Cleveland

-RRB-

of

the

program

S-PLUS

-LRB-

MathSoft

-RRB-

where

Acknowledgments

This

work

is

supported

by

grants

from

the

National

Institutes

of

Health

R.L.D.

is

an

NIH

Post-doctoral

Fellow

Downloaded

from

on

May

Published

by

Cold

Spring

Harbor

Laboratory

Press

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