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