SHAPES AND DYNAMICS OF BIOLOGICAL SYSTEMS

Patrice KOEHL

Department of Computer Science Genome Center

UC Davis

Biology = Quantitative Science ….

Comparing (biological) shapesPart I: Optimal diffeomorphism

Part II: Geodesics in shape spaceTf = 0.001

-2 -1-0.5

0

0.5

1

1.5

2

Time ×10-30 0.2 0.4 0.6 0.8 1

Ener

gy

-150

-100

-50

0

50Tf = 0.001

Tf = 0.07

-2 -1-0.5

0

0.5

1

1.5

2

Time

Ener

gy

-150

-100

-50

0

50Tf = 0.07

Tf = 0.5

-2 -1-0.5

0

0.5

1

1.5

2

Time

Ener

gy

-150

-100

-50

0

50Tf = 0.5

0 1 0 1

0 0.02 0.04 0.06

0 1

0 0.1 0.2 0.3 0.4 0.5

M1

M2

M3

M1

M2

M3

M1

M2

M3

We want to compare two surfaces by finding an optimal diffeomorphism between them.

If the surfaces have identical geometry then the optimal diffeomorphism is given by an isometry.

But what if they have different geometries?

What map is closest to being an isometry?

Optimal diffeomorphims

Optimal diffeomorphisms do more than give a distance. They also give a correspondence.

Optimal diffeomorphims

Diffeomorphic mapping

➢General maps between two surfaces deform lengths and angles

➢Isometries conserve lengths and angles…. but they are rarely appropriate

➢Conformal maps are the next best options, as they distort lengths but preserve angle

The Uniformization TheoremTheorem [Poincaré, Koebe]

Any two genus-zero surfaces are conformally equivalent

Given any two shapes (with no holes), there is a map from one to the other that preserves angles.

The UC Davis Version…

The UC Davis Version…

€

˜ g = e2ug

€

g

€

˜ g

Mapping Genus 0 Surfaces to the Sphere

€

˜ g = e2ug

€

g

€

˜ g

Mapping Genus 0 Surfaces to the Sphere

g: E -> R+ (i,j) -> lij

g’: E -> R+ (i,j) -> l’ij

lij' = eu(i)+u( j )lij

Continuous:

Discrete:

Many algorithms exist:

1. Discrete Ricci Flow 2. Discrete Yamabe Flow 3. Conformal Mean Curvature Flow 4. Harmonic Maps 5. Finite Elements 6. Optimize a cost function 7. Discrete Differential Equation 8. Wilmore Flow 9. Circle Packings

Mapping Genus 0 Surfaces to the Sphere

Many algorithms exist:

1. Discrete Ricci Flow 2. Discrete Yamabe Flow 3. Conformal Mean Curvature Flow 4. Harmonic Maps 5. Finite Elements 6. Optimize a cost function 7. Discrete Differential Equation 8. Wilmore Flow 9. Circle Packings

Mapping Genus 0 Surfaces to the Sphere

(Springborn et al, 2008)

Parametrizing a conformal map between two surfaces

??

Parametrizing a conformal map between two surfaces

m(z) = az+bcz+d

Parametrizing a conformal map between two surfaces

m(z) = az+bcz+d

Discrete stretching energy

How round are proteins?

How round are proteins?

Analysis of anatomical dataDataset of proximal first metatarsals from 38 prosimian primates, and 23 New and Old World monkeys

Prosimian: lemur

Simian: White eared titi (new world)

Simian: Cape baboon (old world)

Analysis of anatomical data

Analysis of anatomical data

Comparing lower molars from primatesRa

te o

f tru

e po

sitiv

es

Rate of true negatives

Comparing lower molars from primates: A10-A13: same order, same family Q06 : same order, different family

A10 0.0 0.38 0.30

Comparing lower molars from primates: A10-A13: same order, same family Q06 : same order, different family

A10 0.0 0.38 0.30 0.36

A10R 0.36 0.18 0.22 0.0

Comparing lower molars from primatesRa

te o

f tru

e po

sitiv

es

Rate of true negatives

Comparing (biological) shapesPart I: Optimal diffeomorphism

Part II: Geodesics in shape spaceTf = 0.001

-2 -1-0.5

0

0.5

1

1.5

2

Time ×10-30 0.2 0.4 0.6 0.8 1

Ener

gy

-150

-100

-50

0

50Tf = 0.001

Tf = 0.07

-2 -1-0.5

0

0.5

1

1.5

2

Time

Ener

gy

-150

-100

-50

0

50Tf = 0.07

Tf = 0.5

-2 -1-0.5

0

0.5

1

1.5

2

Time

Ener

gy

-150

-100

-50

0

50Tf = 0.5

0 1 0 1

0 0.02 0.04 0.06

0 1

0 0.1 0.2 0.3 0.4 0.5

M1

M2

M3

M1

M2

M3

M1

M2

M3

Minimum Action Paths and Shape Similarity

1. Defining a (geodesic) distance between shapes

2. Applications to simple 2D potentials

3. Applications to proteins: a simplified potential

4. Applications to large shapes: more simplifications

D = x1 − x2( )2 + y1 − y2( )2

M1

M2

Distance between Shapes…

M1

M2

D = x1 − x2( )2 + y1 − y2( )2

M1

M2

€

∂L∂X

=ddt

∂L

∂X.

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

M1

M2

L = dXdt

⎛

⎝⎜

⎞

⎠⎟2

D = x1 − x2( )2 + y1 − y2( )2

M1

M2

M1

M2

€

∂L∂X

=ddt

∂L

∂X.

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

M1

M2

L = dXdt

⎛

⎝⎜

⎞

⎠⎟2

S = Ldt0

T

∫ = x1 − x2( )2 + y1 − y2( )2

D = x1 − x2( )2 + y1 − y2( )2

M1

M2

M1

M2

M1

M2

€

∂L∂X

=ddt

∂L

∂X.

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

M1

M2

L = dXdt

⎛

⎝⎜

⎞

⎠⎟2

S = Ldt0

T

∫ = x1 − x2( )2 + y1 − y2( )2

D = x1 − x2( )2 + y1 − y2( )2

M1

M2

Diffusive form of Langevin Equation:η: friction M: diagonal mass matrix U: potential energy B: random force

For a trajectory

where the action, S, is given by (Onsager and Machlup, 1953):

X(0)→ X(t)→ X f (Tf )

P X(0)→ X f (Tf )( )∝ exp −SkBT

⎛

⎝⎜

⎞

⎠⎟

L = ηM dXdt

+∇U(X)⎛

⎝⎜

⎞

⎠⎟2

Corresponding Lagrangian:

Let M be a (smooth) manifold and

E a function from (M,TM) το [0,∞);

Let X0 and XF be two points on M. Then

d(X,Y ) = inf E(γ (t),γ•

(t))dt γ ∈C1([0,T ],M ),γ (0) = X0,γ (T ) = XF0

T

∫⎧⎨⎩

⎫⎬⎭

defines an intrinsic quasi-metric on M.

There always exist length minimizing curves on (M,E). Such curves can always be reparametrized to be geodesics, and any geodesic must satisfy the Euler-Lagrange equation for F(γ):

F(γ ) = L(γ (t),γ•

(t))X0

XF

∫ dt = E(γ (t),γ•

(t))⎡⎣⎢

⎤⎦⎥X0

XF

∫2

dt

There always exist length minimizing curves on (M,E). Such curves can always be reparametrized to be geodesics, and any geodesic must satisfy the Euler-Lagrange equation for F(γ):

F(γ ) = L(γ (t),γ•

(t))X0

XF

∫ dt = E(γ (t),γ•

(t))⎡⎣⎢

⎤⎦⎥X0

XF

∫2

dt

Onsager and Machlup (1953) action:

L = ηM dXdt

+∇U(X)⎛

⎝⎜

⎞

⎠⎟2

Corresponding Lagrangian:

Lagrangian:

Euler Lagrange equations: ∂L∂X

=ddt

∂L

∂X•

⎛

⎝⎜⎜

⎞

⎠⎟⎟

Boundary conditions:

X(0) = X0 X(Tf ) = X f

d 2Xdt2

=∇∇U(X)∇U(X)

L = ηM dXdt

+∇U(X)⎛

⎝⎜

⎞

⎠⎟2

Minimum Action Paths and Shape Similarity

1. Defining a (geodesic) distance between shapes

2. Applications to simple 2D potentials

3. Applications to proteins: a simplified potential

4. Applications to large shapes: more simplifications

Tf = 0.001

-2 -1-0.5

0

0.5

1

1.5

2

Time ×10-30 0.2 0.4 0.6 0.8 1

Ener

gy

-150

-100

-50

0

50Tf = 0.001

Tf = 0.07

-2 -1-0.5

0

0.5

1

1.5

2

Time

Ener

gy

-150

-100

-50

0

50Tf = 0.07

Tf = 0.5

-2 -1-0.5

0

0.5

1

1.5

2

TimeEn

ergy

-150

-100

-50

0

50Tf = 0.5

0 1 0 1

0 0.02 0.04 0.06

0 1

0 0.1 0.2 0.3 0.4 0.5

M1

M2

M3

M1

M2

M3

M1

M2

M3

Minimum Action Paths and Shape Similarity

1. Defining a (geodesic) distance between shapes

2. Applications to simple 2D potentials

3. Applications to proteins: a simplified potential

4. Applications to large shapes: more simplifications

Elastic network for biomolecules:

Elastic potential:

2nd order Taylor expansion:

Elastic network for biomolecules:

V0 (X) =12X − X0( )T H (X0 ) X − X0( )

VF (X) =12X − XF( )T H (XF ) X − XF( )

Mixing potential:

U(X) = − log e−V0 (X ) + e−VF (X )( )

Mixing potential for transition path:

with boundary conditions:

X(0) = X0 X(Tf ) = X f

d 2Xdt2

=∇∇U(X)∇U(X)

Solve:

using a relaxation method.

Transition Path for a Ribonuclease III

Minimum Action Paths and Shape Similarity

1. Defining a (geodesic) distance between shapes

2. Applications to simple 2D potentials

3. Applications to proteins: a simplified potential

4. Applications to large shapes: more simplifications

€

U1(X) = dij (X) − dij (X0)( )2

edges( i, j )∑

An elastic model for shapes

V0 (X) =12X − X0( )T H (X0 ) X − X0( )

VF (X) =12X − XF( )T H (XF ) X − XF( )

Mixing potential:

Mixing potential for transition path:

U(X) =min V0 (X),V1(X)( )

€

˙ ̇ X = H(X0)T H(X0)(X − X0) t ≤ t0

˙ ̇ X = P(X f )T P(X f )(X − X f ) t ≥ t0

⎧ ⎨ ⎩

€

∂L∂X

=ddt

∂L

∂X.

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Euler-Lagrange equations for stationary action:

€

X(t → t0) = Xt = X(t0← t)˙ X (t → t0) = ˙ X (t0← t)

U(X(t → t0)) = U(X(t0← t))

€

X(t) = sinh(Ht)csch(Ht0)(Xt − X0) + X0

€

X(t) = sinh(P(t −T))csch(P(T − t0))(X f − Xt ) + X f

X(0)=X0 X(T)=Xf

X(t0)=Xt

Transitions between two states

Analysis of anatomical dataDataset of proximal first metatarsals from 38 prosimian primates, and 23 New and Old World monkeys

Prosimian: lemur

Simian: White eared titi (new world)

Simian: Cape baboon (old world)

0 10 20 30 40 50Time

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Energy

01

02

03

0 10 20 30 40 50Time

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Energy

01

02

03

04

40 50

0 10 20 30 40 50Time

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Energy

02

0 10 20 30 40 50Time

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Energy

0 10 40 50

Baboon A Lemur AS

urf

ace

BS

urf

ace

A

dact=0.08 dact=0.71

dact=0.26 dact=0.03

0 10 20 30 40 50Time

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Energy

01

02

03

0 10 20 30 40 50Time

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Energy

01

02

03

04

40 50

0 10 20 30 40 50Time

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Energy

02

0 10 20 30 40 50Time

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Energy

0 10 40 50

Baboon A Lemur AS

urf

ace

BS

urf

ace

A

dact=0.08 dact=0.71

dact=0.26 dact=0.03

Bab

oon

Lem

ur

10 20 30 40 50 60

10

20

30

40

50

60

Simian Prosimian

Simian

Prosimian

Observer 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10 20 30 40 50 60

10

20

30

40

50

60

Simian

Prosimian

Simian Prosimian

Observer 2

10 20 30 40 50 60

10

20

30

40

50

60

Simian

Prosimian

Simian Prosimian

Action

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10 20 30 40 50 60

10

20

30

40

50

60

Simian Prosimian

Simian

Prosimian

Geometry

Human01

Human03

Human08

Human07

Human02

Human05

Human06

Human09

Human10

Human0

4

Gorilla01

Gorilla02

Gorilla04

Gorilla05

Gorilla09Gorilla08Gorilla03G

orilla06

Gorilla07

Orang01

Orang02

Orang03

Orang09

Orang07

Orang06

Orang08

Orang10O

rang04

Orang05

Chimp04

Chimp03Ch

imp07

Chimp11

Chimp05

Chimp06

Chimp09

Chimp10

Chimp08

Chimp12

Chimp02

Bonobo04

Bonobo02Bonobo01

Bonobo03

Chimp01

HO

MO

GO

RIL

LA

PONGO

PA

N

Bonobo01

Bonobo03

Bonobo02

Chimp12

Bonobo04

Chimp01

Chimp09

Chimp10

Chimp07

Orang0

5

Chimp11

Chimp02

Chimp03

Chimp06

Gorilla07Gorilla04Gorilla05G

orilla01

Gorilla08

Gorilla09

Gorilla02

Chimp05

Chimp08

Orang06

Orang08

Orang01

Orang02

Orang07

Orang03

Orang09Or

ang04Ch

imp04

Orang1

0

Gorilla03

Gorilla06

Human01

Human03

Human08

Human10

Human09

Human02

Human05

Human06

Human04

Human07

HO

MO

PAN

GO

R

ILLA

PONGO

A) Observer distance B) Variational distanceTrained morphometrist Variational distance

Marc Delarue, Institut Pasteur, Paris Henri Orland, CEA, Saclay

Herbert Edelsbrunner (IST Austria), Seb Doniach (Stanford University)

Michael Levitt, Stanford U. Joel Hass, UC Davis

Funding: NIH, NSF, Sloan Foundation, NU Singapore

Thank You