Structural Complexity of Life is the Hierarchy of Devices

1

Many thanks to Weishi

For so many good things

For many years

Glad to be back in Lawrence in the (near) spring!





u

2

Structural Complexity

of Life

is the

3

4

Voltage Sensor

Conduction

Pore

Gating Structure

Conduction

5

Biology goes from PARTICULAR to the General

What can we learn

from the

Voltage Sensor

About

6

Real Biological System

A Nerve Cell is a Hierarchy of Devices

Cell Body,

Dendrites,

Axon,

Terminals

Example:

Axon

8

Classical cable theory of transmission lines, telegrapher’s equations, Kelvin, Hodgkin,

Noble,

including 3D-cable theory, ~10 papers, e.g.,

Barcilon, Cole, Eisenberg. 1971. Singular Perturbation … SIAM J. Appl. Math. 21:339-354.

Page 9

How do ions move through channels?

About 200 papers since 1986

Today,

How does Voltage tell a Channel to

open?

One paper, so far

Page 11

Page 12

Nerve Signaling

is a

I287 S240 F244 I237 I241 R1 R3 R2 R4 F290 C286 V236

S1

S2

S3

S4

S1

S2

S3

V236I237

S240

I241

F244 C286I287F290 A319I320

Shaker VILLSIVIFCLE ETLCIIWFTFE DIIAIIP Kv1.1 VILISIVIFCLE ETLCIIWFSFE DIVAIIP Kv1.2 VILISIVSFCLE ETLCIIWFSFE DIVAIIP Kv2.1 FIVLSTIALSLN EAVCIAWFTME DLLAILP Kv3.1 FILVSITTFCLE EGVCVVWFTFE DFVAILP Kv4.1 FIAVSVIANVVE DTACVLIFTGE DVVAILP Kv5.1 LILVSSVVMCMG ETACIGWFTLE DVLAILP Kv6.4 FVIVSMVVLCAS EAICIGWFTAE DLLAITP Kv7.1 IVLVCLIFSVLS EIVLVVFFGTE DLIVVVA KvAP AALLSVIVVVVE DLILVIILWAD EIPALVP Nv11D4 LICLNMVTMMVE NLVFIVLFTGE DFVVVIL Nv14D1 FIMITILTNCVF EYTFTGIYTFE DFSVITM Nv15D3 FIIFMILLSSGA DKMFTYVFVLE DFLIVDV Nv17D2 AITICIVLNTLF NLVFTGIFAAE DSLIVTL Ci-VSP LIFLDIILMIID ALALSCYFMLD DGLIIVV Hv1 LVVLDALLVLAE SITILVFFMME DAVVVVV

Shaker Kv1.1 Kv1.2 Kv2.1 Kv3.1 Kv4.1 Kv5.1 Kv6.4 Kv7.1 KvAP (DIV) Nav1.1 (DI) Nav1.4 (DIII) Nav1.5 (DII) Nav1.7 Ci-VSP Hv1

We studied

gating currents

after exchanging

each

of the

10 side chains by the

series

of amino acids.

Summary of results

Lacroix et al, identified specific

physico-chemical principles by which the gating

charges are transferred across a Kv

channel voltage-sensor

.

●A steric gap at position

S240

defines

the primary entryway for the gating

charges during activation and extends

the field to be sensed by the first

charge

●A steric/hydrophobic barrier located

at position

C286

and

I287

tunes the

main energy barrier for the gating

transition.

Lacroix, Hyde,Campos, and

Bezanilla

,

Proc Natl Acad Sci U S A, 2014.

Vol. 111 p.E1950

16

Plug = Push Rod = S4

In Gating Structure

Not in Conduction Pore

Conduction

Pore

Gating Structure

Conduction

I287 S240 F244 I237 I241 R1 R3 R2 R4 F290 C286 V236

S1

S2

S3

S4

intracellular

extracellul

Voltage Sensor

works by

Charge Injection

19

Interdisciplinary Problem Distinguished Biologists§ call a Saturating Distribution

‘Boltzmann’ e.g.,

Physicists: Saturation Fermi distribution

Boltzmann* distribution does NOT saturate.

Boltzmann is exponential, like.

*Boltzmann (1904) Lectures on Gas Theory, Berkeley

§ _{p.503 of Hodgkin and Huxley. 1952.}

‘Quantitative description ...’ J. Physiol. 117:500-544. Bezanilla, Villalba-Galea J. Gen. Physiol (2013) 142: 575

Bezanilla. How membrane proteins sense voltage Nature Rev Mol Cell Biol (2008) 9, 323

Fermi

Distribution

not Boltzmann

Arginines

Internal Dissolved Ions

External Dissolved Ions

Dissolved Ions External Dissolved Ions

Voltage Clamp Test Potential Voltage Clamp =0 Holding Potential Current

Perhaps the first

Consistent Model of a Protein Machine

21

Francisco Bezanilla

Chun Liu

柳 春

Allen Tzyy-Leng Horng

23

𝐸 =

න ቈ

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𝑇 ෍

𝑐

𝑙𝑜𝑔 𝑐

−

𝜀

𝜀

2

𝑎𝑙𝑙 𝑖

ȁ̏

∇𝜙 ȁ̏

+ ෍

𝑧

𝑒

𝑎𝑙𝑙 𝑖

𝑐

𝜙 + ෍

ሺ

𝑉

+ 𝑉

ሻ

𝑎𝑟𝑔 𝑖𝑛𝑖𝑛𝑒𝑠

𝑐

𝑉

+ ෍

𝑔

2

𝑐

𝑐

𝑎𝑟𝑔 𝑖𝑛𝑖𝑛𝑒𝑠 𝑖,𝑗

෍ 𝑑𝑉,

Variational Formulation

EnVarA

because

Everything interacts with Everything Else

Poisson Equation and Transport equation are DERIVED from variations like

30

Defining Laws

Charge Creates Electric Field

−

1

𝐴

𝑑

𝑑𝑧

(

Γ

𝐴 𝑑 𝜙

𝑑𝑧

)

=

∑

𝑖

=

1

𝑁

𝑧

𝑖

𝑐

𝑖

,

𝑖

=

Na

, 

Cl

, 1, 2, 3, 4

Transport of Mass

25

where

K

is the spring constant,

z

is the fixed anchoring position of the spring for each arginine

c

on

S4

, is the center-of-mass

z

position of

S4

by treating

S4

as a rigid body.

follows the motion of equation based on spring-mass system:

Current Carried by Arginines

note cross terms

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Page 27

from -90mV to VmV at t=10, holds on till t=150, and drops back to -90mV, where V=-62, -50, … -8 mV. (b) τ2 versus V compared with experiment [7].

Page 28

and arginines at (b) V=-90mV, (c) V=-48mV, (d) V=-8mV.

29

Ions

Electric

Field

Current is Conserved

OUTPUT Current including INPUT Voltage Clamp

t

Conservation of Current is an

Important Constraint

.

Rate Models of Chemical Kinetics;

Molecular Dynamics

do

not

conserve current

INPUT

Voltage Clamp

INPUT

Voltage Clamp

INPUT

Voltage Clamp

INPUT

Voltage Clamp

34

End of Particular Story

35

General Story

36

Science goes from PARTICULAR to the General

Page 37

How can a few thousand atoms

Page 38

How can a few thousand atoms

conceivably control 10

25

atoms?

T

he thousand atoms of one gene occupy say 10

m

The volume of a person might be 1m

Volume of USA

China 1m high is 10

m

Fraction of space of a gene is about 10

-27

Fraction of Space of One Person in USA is 10

-13

1 m

3

has no effect in USA

39

A few atoms make a

BIG Difference

Current Voltage relation determined by

John Tang

in Bob Eisenberg’s Lab

Ompf

G

119

D

Glycine G

replaced by

Aspartate D

OmpF

1M/1M

G119D

1M/1M

G119D

0.05M/0.05M OmpF

0.05M/0.05M

Structure determined by

Raimund Dutzler

Page 40

Life is special because it is

inherited from a tiny number

of atoms

And the central question of biology is

Page 42

How can a few thousand atoms

conceivably control 10

atoms?

Traditional Statistical Mechanics says this is impossible!

where and

specifies the radius of the small spherical volume over which the

spatial average takes place.

Page 43

How can a few thousand atoms

conceivably control 10

atoms?

Biological Answer:

Structure: a Hierarchy of Devices

Physical Answer:

Electrodynamics: Strong and Universal

inside atoms to stars

Another talk*

another day!

*Eisenberg, Oriols, and Ferry. 2017. Dynamics of Current, Charge, and Mass. Molecular Based Mathematical Biology 5:78-115

Everyone knows Biology

is made of

Structures

Working hypothesis:

The Structures

make

Devices

that span the scales

What is

a Dev

ice?

How a

re the

y stud

ied?

A diffe

rent ta

lk!

Page 45

Different Kind of Averaging

in Device

Definition of a Device

Output is Perfectly Correlated with Input

Averaging in a Device Creates a Perfectly Correlated Replica of the Input

Not

equal

Precise

stochastic

definition

Devices are often Branched One Dimensional Circuits because

Maxwell Equations Imply Perfect Correlation of Currents

Page 46

Structural Complexity

so characteristic of life,

so daunting to mathematicians

is the

Hierarchy of Devices

What is

a Dev

ice?

How a

re the

y

studie

d?

A diffe

Page 47

How can a few thousand atoms

conceivably control 10

25

atoms?

ANSWER:

by forming a

Page 48

Nerve Signaling

is a

Voltage Sensor

One Device

in the Hierarchy that makes a Nerve

49

50

51

52

Plug = Push Rod = S4

In Gating Structure

Not in Conduction Pore

Conduction

Pore

Gating Structure

Conduction

Page 53

54

Classical cable theory of transmission lines, telegrapher’s equations, Kelvin, Hodgkin,

Noble,

including 3D-cable theory, ~10 papers, e.g.,

Barcilon, Cole, Eisenberg. 1971. Singular Perturbation … SIAM J. Appl. Math. 21:339-354.

Device converts Input to Output by a simple ‘law’

55

Device is ROBUST and TRANSFERRABLE

because it uses POWER and has complexity!

Dotted lines outline: current mirrors (red);differential amplifiers (blue);

class A gain stage (magenta); voltage level shifter (green);output stage (cyan).

Circuit Diagram of common 741 op-amp: Twenty transistors needed to make linear robust device

INPUT

V

(t)

OUTPUT

V

(t)

Power Supply

Dirichlet Boundary Condition

independent of time

and everything else

Integrated Circuit

Technology as of ~2014

IBM Power8

56

58

Muscle

Skeletal

Nerve Muscle

Synapse

59

One Cell

contains many

Devices

Structural Complexity of Life

is the

Hierarchy of Devices

60

Mathematics

describes only a tiny part of life,

But

Mathematics* Creates

our

Standard of Living

*e.g.,

Electricity, Computers, Fluid Dynamics, Optics, Structural Mechanics, ….

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u

61

How can we use mathematics to describe

biological systems?

I believe some biology is

Physics ‘as usual’

‘Guess

and

Check’

But you have to know which biology!

Anoth

er tal

k,

anoth

er da

y

“Scie

nce a

s a

Sequ

ence

of

Inver

se P

63

Classical cable theory of transmission lines, telegrapher’s equations, Kelvin, Hodgkin,

Noble,

including 3D-cable theory, ~10 papers, e.g.,

Barcilon, Cole, Eisenberg. 1971. Singular Perturbation … SIAM J. Appl. Math. 21:339-354.

Page 64

Channels are Source of Signal

Lipid molecule

Page 66

Biology is made of

Devices

and they are Multiscale

Structural Complexity

of Life

is the

67

Vargas, E., Yarov-Yarovoy, V., Khalili-Araghi, F., Catterall, W. A., Klein, M. L., Tarek, M., Lindahl, E., Schulten, K., Perozo, E., Bezanilla, F. & Roux, B.

An emerging consensus on voltage-dependent gating from computational modeling and molecular dynamics simulations.

The Journal of General Physiology 140, 587-594 (2012).

Emerging Consensus ….

Voltage Sensor Structure

intracellular

extracellul

Voltage Sensor

works by

Charge Injection