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20.GEM GEM4 Summer School: Cell and Molecular Biomechanics in Medicine: Cancer
Summer 2007
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Space, Time and Energy Landscapes
related to Life
Ju Li
Courtesy of Ju Li. Used with permission.
2nd GEM4 Summer School
Cell and Molecular Mechanics in BioMedicine
with a focus on Cancer
June 25– July 6, 2007
National University of Singapore
Outline
• Earth as Heat Engine, Life as Information • Energy Scales
• Spatial Pattern of Electrostatic Energy
• Spatial-Temporal Characteristics of Bending
• Rare Events and Timescale
2
Cosmic radiation
background: 2.7K
5800K
hig
hen qua
erg li
y i ty
n
low-quality energy out
Images courtesy of NASA.
Carnot ideal
heat engine efficiency:
(T1-T2)/T1
In terms of quantity: Ein ≈ Eout
But in terms of quality or free energy,
earth enjoys (spends) huge drop 3
1
Energy (eV)
2
High-quality energy in
3
4
Solar radiation
(photons/area/time)
Low-quality energy out
The Earth's Infrared Thermal Emission
Intensity
280k
O3
6.2
7.1
8.3
10.0
-6
10 Meters
CO2
12.5
16.7
25.0
Wavelength
Figure by MIT OpenCourseWare.
Do Thermodynamics Govern Life?
Equilibrium: given the constraints, the macro-condition
of system that is approached after sufficiently long time
Non-equilibrium
states
Ar gas atoms
Now stir it
any way
you like
Vacuum
outside
insulating box
(the constraint)
Equilibrium system → T, S
and wait …
Equilibrium
state
Vacuum
outside
Non-equilibrium system: no T, S
5
Probability
equilibrium Maxwellian distribution
ions in a
Tokamak plasma:
non-equilibrium
 m | v − v 0 |2 
d 3v
exp  −

3/ 2
(2π k BT / m )
2
k
T
B


∝ kBT
bimodal distribution:
no temperature!
v0
atom velocity
6
equilibrium is however yet a bit more subtle:
It is possible to reach equilibrium among a subset
of the degrees of freedom
(all atoms in a shot) or subsystem, while this subsystem is not in equilibrium with the rest of the system.
Image removed due to copyright restrictions.
Photograph of athlete catching a ball.
This is why engineering &
material thermodynamics is useful for cars and airplanes. 7
thermodynamics could
apply to individual
components, but not the
entire machine.
Image removed due to copyright restrictions.
Simple schematic of an automobile.
Life on earth are soft carbonaceous machines, that consume
high-quality energy (metabolism) to achieve functions impossible from totally equilibrium systems.
But biomachines differ from cars and airplanes in one aspect: the dominance of information.
8
Fact: in few months, 90% of my physical self (atoms) becomes CO2, urine, etc. Life as solitons. Soliton is a nonlinear local excitation (energy pack) that conserves character (non-dispersive) when traveling in a medium.
energy density
two-soliton collision in cubic nonlinear Schrödinger system
time
space
9
Life is a very specific program stored in DNA to
consume free energy in the physical world. (acquire and spend)
I caught a rabbit and ate it. One week later, the atoms that caught the
rabbit become CO2, while some of the rabbit’s atoms gets incorporated
into new “me”. Why “I ate the rabbit”, instead of “the rabbit ate me”?
Information triumph over matter
Image removed due to copyright restrictions.
Cover of textbook: Nelson, Philip, Marko Radosavljevic, and Sarina Bromberg.
Biological Physics: Energy, Information, Life. New York, NY: W. H. Freeman and Co., 2004.
ISBN: 9780716743729.
Life is information
organizing energy.
10
Courtesy of American Institute of Physics. Used with permission.
Rob Phillips and Stephen R. Quake, “The biological frontier of physics,” Physics Today 59 (2006) 38.
11
Subra Suresh, “Biomechanics and biophysics of
cancer cells,” Acta Materialia 55 (2007) 3989.
Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission.
12
Last philosophical question: which are more magical, biomachines or engineered machines ? Biomachines are severely confined in energy scale.
Engineers: No bird flies faster than the plane I took to Singapore
No animal went to the Moon with bio power
No laser in bio-systems, no nuclear reactor
No purified silicon to augment computation
Biologists: answer 1: efficiency, complexity, adaptivity
of ATP burning soft nano-machines
answer 2: All hard machines are made by one self-aware soft machine, Cro-Magnon.
Modelers: modeling is the ultimate self-awareness.
Correct answer: the most potent magic belongs to nano-bio-engineers who know how to do modeling! (information + energy scales)
13
Spatial Organization of Electrostatic Energy
+e
r = 1Å
vacuum
-e
q1q2
U=
= 14.4 eV
4πε 0 r
This is huge! (k BTroom = 0.025 eV, U = 560k BTroom )
Now add water:
q1q2
14.4
U=
= 0.18 eV = 7k BTroom
=
4πε rε 0 r
80

U 
−3
exp  −
10
=

 k BTroom 
this is good for life
14
Screening by bound charge vs Screening by mobile charge:
Screening by bound charges reduces the magnitude
of the long-range electric field but can never kill it.
Screening by mobile charges completely kills the long-range tail of electric field.
Poisson's equation in dielectric medium:
pH-7 water is strong
dielectric but weak electrolyte:
[hydronium]=[hydroxyl]=10-7
ρ total
∇φ=− ε rε 0
2
electrolyte
ρ total = eδ ( x ) + Zeρ p-ion (x ) − Zeρ n-ion (x )
Define φ ( x → ∞) = 0, we know also that
ρ p-ion (x → ∞) = ρ n-ion (x → ∞) = ρ 0
15
At x ≠ ∞, p-ion has energy Zeφ ( x ), and
n-ion has energy − Zeφ ( x ) compared to at infinity,
So if thermal equilibiurm is reached,  Zeφ ( x ) 
ρ p-ion (x ) = ρ 0 exp  −
,
k BT 

 Zeφ ( x ) 
ρ n-ion (x ) = ρ 0 exp 

 k BT 
 Zeφ (x) 
ρ total = eδ ( x ) − 2Zeρ 0 sinh 

 k BT 
eδ ( x ) − 2Zeρ 0 sinh( Zeφ / k BT )
2
∇φ=− : nonlinear
ε rε 0
16
Suppose Zeφ ( x) << k BT , linearize the equation:
ρ total
2Z e ρ 0
≈ eδ ( x ) −
φ (x)
k BT
∇φ=−
2
2 2
eδ ( x )
ε rε 0
2Z e ρ 0
+
φ (x)
k BT ε rε 0
2 2
2 2
eδ ( x ) φ
2Z
e ρ0
-2
2
+ 2
λD ≡
: ∇φ=−
ε rε 0
λD
k BT ε rε 0
exp( − r / λD )
φ=
4πε rε 0
r
e
17
For water solvent (ε r =80) at 298 K:
0.3nm M
1
2
, I ≡ ∑ Z i ci
λD =
2 i
I
ionic
strength [M]
mobile ion
concentration [M]
λD is how much free charges can be separated (local breakdown
of electro-neutrality) spatially by thermal fluctuation energy.
Unless you are doing Van de Graaff particle accelerator where
MeV energy are involved
The most you can do in ordinary
Image removed due to copyright restrictions.
Van der Graaff generator.
electrochemical system is to separate a
bit of charge (∝area) λD (typically few nms)
away from the balancing counter ions. 18
Spatial-Temporal Characteristics of Uniaxial Compression
Eε xx2
E
2
Strain energy U = ∫ dxdA
= A∫ dx ( ∂ x
u )
2
2
Kinetic energy K = A∫ dx
ρ (∂ t u)2
2
Aρ∂ t2u = AE∂
2
x u
Plug in u( x, t ) = exp(ikx − iωt ), ω 2 =
E
ρ
k 2
frequency ω
slope = speed of sound =
E
ρ
wavevector k
19
Spatial-Temporal Characteristics of Bending
Bending is special
neutral plane:
length is
preserved
2
Eε xx
Ey 2
Strain energy U = ∫ dxdA
= ∫ dxdA 2
2
2R
κC2
1
EI
Moment of inertia: I ≡ ∫ y dA : U = ∫ dx 2 = ∫ dx
, κ ≡ EI , C ≡
2R
2
R
1 2
π a 4
for cylindrical rod of radius a : I = ∫ r 2π rdr =
4
2
2
20
1 d 2
y
C≡ = 2
R dx
( y − R) + x = R
2
2
2
y = R − R2 − x2
2
x
=
+ ...
2R
21
Spatial-Temporal Characteristics of Bending
Bending is special
κC2
EI 2 2
Strain energy U = ∫ dx
= ∫ dx
∂x y)
(
2
2
ρ (∂ t y)2
Kinetic energy K = A∫ dx
2
Aρ∂ t2 y = EI ∂ 4x y
Plug in u( x, t ) = exp(ikx − iωt ), ω 2 =
EI 4
k
ρA
22
frequency ω
∝ k2
wavevector k
Long-wavelength free bending waves are extra-soft.
σ Ak 2 + EIk 4
If tension σ is applied, ω =
ρA
2
If compression σ is applied, a long enough rod always buckle:
-1
∝
buckling wavelength ∝ k min
EI
: Euler buckling formula
σA
23
Consider a quarter-circle: 2π R κ
πκ
Energy required:
=
∼ k BT
2
4 2R
4R
R
Smallest loop radius thermal fluctuation can bend: R ∼
πκ
4k BT
Persistence length: longest separation that orientation correlation can be survive 24
Rare Events and Timescale: Harmonic Transition State Theory
3N
Potential energy landscape: U (x )
∂U
Force = − 3N
∂x
2
∂U
Hessian =
3N
3N
∂x ∂x
N is total number of atoms
Diagonalize Hessian to get normal mode frequencies
ν 1 ,ν 2 ,ν 3 ,...,ν 3N
local minima: ν 1 ,ν 2 ,ν 3 ,...,ν 3N are all real
saddle point: ν 2 ,ν 3 ,...,ν 3N are real
25
Maximum
Minimum
Saddle
ν 1ν 2ν 3...ν 3N
 U′ −U 
R=
exp  −

ν 2′ν 3′...ν 3N ′
 k BT 
 F′ − F 
= ν 1 exp  −
 , where F ′ = U ′ + k BT lnν 2′ν 3′...ν 3N
′
 k BT 
F ′ = U + k BT lnν 2ν 3 ...ν 3N
26
In quantum mechanics, free energy of one oscillator is
− hν/2k BT
e
−k BT ln(e − hν/2kBT + e −3hν/2 kBT + e −5hν/2 kBT + ...) = −k BT ln
−hν / k BT
1− e
hν
In the semiclassical limit (h → 0), F → k BT ln
k BT
basin B
basin A
Within certain timescales, it is possible to reach equilibrium
among states within Basin A (steady-state probabilities in
master equation), but not with respect to states of Basin B.
27