Strings, Integrable Models and Biology:�

“The problem of life is one of the three fundamental problems in theoretical physics” (Dirac 1931)

A New Frontier? M.Chernodub, U..Danielsson, S.Hu, Y. Jiang, D. Jin, A.Krokhotin, A. Liwo, M.Lundgren, N.Molkenthin, X.Peng, H. Scheraga, …

http://www.folding-protein.org

A string-like object in three dimensions has four different phases:

Order parameter = end-to-end distance

Rg = R0L⌫(1 +

R1

L�+ . . . )

⌫ =

8>><

>>:

1 di↵erentiable curve

3/5 self � avoiding random walk

1/2 ordinary random walk

1/3 collapsed string

Four universality classes

Example: Quantum Bosonic string S =

Z LZ

0

d2� gab@aX · @bX

< X(0) ·X(L) >/ L ) ⌫ = 1/2Two-point function:

(L ! 1)

⌫ = inverse Hausdor↵ (fractal) dimension

R2g = R2 ⇠

X

n,m

< rn · rm >

< rn · rm >= 0 for n 6= m

For random walk:

) R2 ⇠X

n

< r2n > ⇠ L

aa2 ) ⌫ = 1/2

Observe: for differentiable curve ⌫ = 1

Universality: < rn · rm >= �mn ⇠ exp{�⇠|n�m|}

ci = rj1 + . . . + rjk

|ci+1 � ci| > ⇠ ) < ci+1ci >! 0

Problem: How to describe collapsed strings

Why interesting? •  New form of (fundamental string) matter •  The Dirac problem of life

•  All metabolic processes in all living cells are proteins in action •  Proteins are string-like objects in the collapsed phase •  Protein Folding Problem : « most important problem in all of Science »

⌫ = 1/3

Extrinsic geometry and energy of strings

Curves and their visualization in three dimensional space: ��In addition of direction, need a frame – think of « rollercoaster »�

x(s) : s 2 [0, L] ! R3 a space curve in R3.

t =1

||x|| x ⌘ 1

||x||dx(s)

ds

unit tangent vector:

Frenet equation:

d

ds

0

@n

b

t

1

A = ||x||

0

@0 ⌧ ��⌧ 0 0 0 0

1

A

0

@n

b

t

1

A

b =x⇥ x

||x⇥ x|| & n = b⇥ t

(s) = ||x⇥x||||x||3 curvature

⌧(s) = (x⇥x)·...x||x⇥x||2 torsion

Frenet frame

(Curve length parameter: ||x|| = 1)

http://demonstrations.wolfram.com/FrenetFrame/

) Frenet frame can not be defined

For a plane curve, inflection point is topological invariant It can not be removed except thru the end points of the curve.

y = � d

dsV (s) = � d

ds

m2

2c2(y2 � c2)2

�= �2m2

c2y(y2 � c2)

y(s) = c · tanh[m(s� s0)]

Special point on a curve: (s0) = 0

Inflection point:

Assume inflection point is simple:

d(s)ds s=s0

6= 0

Inflection point is like a soliton: �

Frame rotation: �✓nb

◆!

✓e1e2

◆=

✓cos ⌘(s) � sin ⌘(s)sin ⌘(s) cos ⌘(s)

◆✓nb

◆

d

ds

0

@e1e2t

1

A=

0

@0 (⌧ � ⌘) � cos ⌘

�(⌧ � ⌘) 0 � sin ⌘ cos ⌘ sin ⌘ 0

1

A

0

@e1e2t

1

A

Rotated Frenet equation: �

T 1 =

0

@0 0 00 0 �10 1 0

1

A T 2 =

0

@0 0 10 0 0�1 0 0

1

A T 3 =

0

@0 �1 01 0 00 0 0

1

AThree dimensional rotation matrices: �

Can write effect of rotation as follows: �

⌧ ! ⌧ � ⌘

T 2 ! (T 2cos ⌘ � T 1

sin ⌘) ⌘ e⌘T3

(T 2) e�⌘T 3

d

ds

�

⇤e1

e2

t

⇥

⌅ =

�

⇤0 (⇤ � �) �⇥ cos �

�(⇤ � �) 0 �⇥ sin �⇥ cos � ⇥ sin � 0

⇥

⌅

�

⇤e1

e2

t

⇥

⌅

• Abelian Higgs multiplet (�±, ⇥) � (⇤, A1)

Essentially unique energy function

Chern-Simons = Chirality

⇥ F =L�

0

(⌅s�)2 + �2⇤2 + ⇥(�2 � a2)2 + b · ⇤

Non-linear Schrodinger Hamiltonian

NLSE is integrable with dark solitons:

Chern-Simons

(2): « Nambu-Goto action » (2)+(4): « Polyakov’s extrinsic string » (4)+(8): « torsion bar »

H�3 = ⌧2

H�2 = ⌧ (1)

H�1 = L (2)

H0 = 0 (3)

H1 = 2(4)

H2 = 2⌧ (5)

H3 = 2s + 2⌧2 + �4

(6)

H4 = modified KdV hamiltonian (7)

(8)

duality???

Can combine into « Hasimoto variable »�⇠(s) = (s) exp

✓i

Z s

0⌧ ds0

◆

Dark soliton of the NLSE:

yss = �dV

ds= �2�2

m2y(y2 �m2) ) y = m tanh[�(s� s0)]

Inflection point is soliton

V (y)

y! c +c0

y(s)

s

!c

+c

s0

0

Proteins and solitons

Proteins have a modular structure SCOP: 1393 folds CATH: 1282 topologies No change since 2008

Data and dynamics suggests that there is some Principle of Collective Self-Organization

Protein folding is coherent and robust at mesoscopic scales

Proposal: •  Proteins are composites of solitons

•  The solitons that describe proteins solve a discrete nonlinear Schrodinger equation ** with sub-atomic precision ***

Discrete Frenet Equation:

bi =ti�1 ⇥ ti

|ti�1 � ti|

ni = bi � ti

Discrete Frenet Equation = Transfer Matrix:

r(s) =s� si

si+1 � siri+1 �

s� si+1

si+1 � siri

ti =ri+1 � ri

|ri+1 � ri|

⇥ rk =k�1�

i=0

|ri+1 � ri| · ti

�

⇤ni+1

bi+1

ti+1

⇥

⌅ = Ri+1,i

�

⇤ni

bi

ti

⇥

⌅

�

⇤ni+1

bi+1

ti+1

⇥

⌅ = Ri+1,i

�

⇤ni

bi

ti

⇥

⌅ � e��i+1,iT2· e�⇥i+1,iT

3

�

⇤ni

bi

ti

⇥

⌅

bi+1 · ti = 0

R =

�

⇤cos ⇥ cos � cos ⇥ sin � � sin ⇥� sin � cos � 0

sin⇥ cos � sin ⇥ sin � cos ⇥

⇥

⌅

cos �i+1,i = ti+1 · ti

cos �i+1,i = bi+1 · bi

discrete Z2 gauge symmetry:

�i+1,i ⇤ �i+1,i � ⇥

⇤k+1,k ⇤ � ⇤k+1,k for all k ⇥ i

Leaves the discrete polygon invariant

*Read PDB data with all bond angles positive *Use gauge symmetry to identify solitons *Angles defined mod(2π)

Landau free energy for folding proteins :

Equation for torsion angle:

E = �N�1⇤

i=1

2 �i+1�i +N⇤

i=1

�2�2

i + c · (�2i �m2)2

⇥

+N⇤

i=1

�b �2

i ⇥2i + d ⇥i + e ⇥2

i

⇥

⇤E

⇤⇥i= 2b�2

i ⇥i + 2e⇥i + d = 0

⇥ ⇥i[�i] = �12

d

e + b�2i

(Generalized) Discrete Nonlinear Schrodinger Equations:

�i+1 � 2�i + �i�1 = U ⇥[�i]�i ⇥dU [�]d�2

i

�i (i = 1, ..., N)

H = �2N�1⇤

i=1

�i+1�i +N⇤

i=1

�2�2

i + U [�i]⇥

Hamiltonian

⇥(n+1)i = ⇥(n)

i � ��

⇥(n)i U ⇥[⇥(n)

i ]� (⇥(n)i+1 � 2⇥(n)

i + ⇥(n)i�1)

⇥

Converges towards solitons

U [�] = �d

4d

e + b�2� 2cm2�2 + c�4

Topological soliton = helix-loop-helix motif

Ansatz for discrete soliton = discretized continuum kink :

�i = (�1)r+1 mr1 · ecr(i�sr) �mr2 · e�cr(i�sr)

2 cosh[cr(i� sr)]

⇥i = �12

br

1 + dr�2i

Result: no more than 200 solitons are needed to cover 92% of PDB with accuracy < 0.7 Angstrom

Example: 1LMB – the paradigm genetic switch

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• Better than B-factor • PicoBiology

Summary: •  Everything in our research suggests that folded proteins can be described in terms of topological solitons in DNLSE • Furthermore, folded proteins can be described in terms of an explicit, universal elementary function • A given soliton structure appears in many different proteins, from eukaryotes to prokaryotes - no more than 200 different profiles describe over 92% of PDB •  For ultra-high resolution proteins we reach accuracies better than 0.2 Angstrom – “picoengineering” with biomolecules is a realistic possibility Question: Could quantum continuum NLSE somehow provide an even wider description of collapsed strings ???