Noises : From Physics to
Biology
卓益忠
2011-10-19
北方温泉会议中心
Crossover Sciences
Mountain in Between
隔行如隔山
To climb over or penetrate
through the mountain between
physics and biology, the noises
play a crucial role: noise assisting
tunneling and climbing.
Part 1,Noise in Physics
1, Phenomenological Theories
(A),布朗运动(Brownian Motion)
(B) 牛顿与朗之万之间（Intermediate
Dynamics between Newton and Langevin)
2, 微观动力学基础(Microscopic Foundation)
Part 2, Noise in Biology
1, Complex dynamics of life at different
scales
(A), Biological response to radiation
(B),主动布朗运动(Active Brownian
motion)
2, New approaches to the solutions of
noisy biological networks
3, Fluctuation-Dissipation Theorem
4, Signal Transduction
5, Perspective
Part 1,Noise in Physics
1,Phenomenological Theories
(A)布朗运动(Brownian Motion)
• 最简单的一种随机行走(random walks)
• 自然界中(物理,化学与生物等)普遍存在
• 广泛应用于原子核大振幅运动(核裂变,
核融合)的输运理论 及各种生物过程。
Brownian Motion phenomenon was
discovered by the Scottish botanist Robert
Brown in 1827.
The first quantitative explanation of
Brownian motion was due to Einstein in 1905.
In his picture the spatial position of a
Brownian particle was a stochastic process.
The beauty of Einstein’s model is that it
permits a quantitative comparison to
experiment.
Einstein’s quantitative explanation for
Brownian motion put the long standing
discussion on the molecular nature of
matter to rest, thus to start a new era for
the science of life.
At the same time, he set the cornerstone
for the development of statistical
mechanics.
Delbrueck et al suggested the molecular
nature of genetic code in 1935.
Schrodinger deduced that the gene was an
aperiodic crystal composed of a linear array
of different isomeric component in his book
What is life(1944)
The structure of DNA was solved by
Watson and Crick in 1953, and thus a new
era of molecular biology started.
 X (t )  6Dt
2
The diffusion constant for spheres of radius
R is given by so-called Stockes-Einstein
formula:
D  kBT / 6 R
Einstein’s picture of Brownian motion
keeps track only of the position of Brownian
particle and ignores the positions and
momenta of all the water molecules, as well
as the momentum of the particle itself.
This, in fact, leads to an unphysical result
for the average velocity of the particle,
pointed out by Einstein.
A simple way to remedy this situation was
taken by Langevin in 1907.Langegevin’s
approach was to consider the mechanical
equation by separating the force into three
parts, in addition to original Newtownian
force, there are viscous forces that tend to
damp the motion of the particle and the
random forces, which are rapid and
unpredictable.
2, 微观动力学基础(Microscopic Foundation)
Dividing the total system into system plus
environment ( reservoir, heat bath) is an
universal approach ( open system):
• 系统＋环境（动力学体系）
H  H s  H e  H coup
( 非线性)

应用投影算符（时间无关与时间有关）
（耦合）主方程
When the environment is time
independent ( heat bath) the time
independent projection operator method is
used:
Projecting on total density matrix
starting from Von Neumann equation
( Schrodinger Representation)
(Nakajima-Zwanzig)
The generalized Master equation is
obtained for density matrix of the system.
Projecting on the dynamical variable for
the total system starting from Heisenberg
equation
(Heisenberg Representation)
( H. Mori)
The generalized Langevin Equation is
obtained.
When environment is time-dependent, the
time-dependent projection operator method
is to be used both in Schrodinger
Representation as well as in Heisenberg
Representation.
The coupled Master equations or the
coupled Langevin-type equations are
obtained.
Due to complication, most derivations are
remained in formulations.
We have been interested in this field for a
long time.
The newest version on this subject by
F.Sakata et al:
The main idea is to develop a sort of
interaction representation (IR) for a complex
system, namely, to treat the irrelevant
subsystem by a density matrix, while the
relevant subsystem by a dynamical
variables.
Part 2, Noise in Biology
1, Complex dynamics of life at different
scales
(A), Biological response to radiation
(B),主动布朗运动(Active Brownian
motion)
2, New approaches to the solutions of
noisy biological networks
3, Fluctuation-Dissipation Theorem
4, Signal Transduction
5, Perspective
1, Complex dynamics of life at different scales
It is not within our reach to accurately model
the entire living cell, or even an entire process
such as the mechanical response or radiation
response.
The so called bottom-up approach is now a
well-accepted method in all of biophysics. The
principal idea is to build models from a limited
number of basic components.
Once these components are experimentally
and theoretically analyzed in detail, the next
ingredient is added until celled-like structures
are created in a controlled function.
As a result of a bottom up approach, deeper
insight into both established mechanisms and
new mechanisms may arise.
A, Biological response to radiation
Exposure of the cells to ionizing radiation
initiates a complex series of physical,
chemical, and biological changes that may
lead to cell death, organ dysfunction, or
cancer.
The responses to such a radiation are at
different scales both at time scale and at
spatial scale.
• From radiation induced DNA damage to the
observable biological endpoint involves many
spatial and temporal scales of different orders
of magnitude.
Central Dogma
DNA
Transcription
Translation
RNA
Protein
Replication
DNA polymerase:
RNA polymerase:
proofreading
without proofreading
Mutation 
Approximations (zero
order)
DNA is assumed to be the key
target as it carries message of life.
DSB is considered to be the
most critical initial damage
(neglecting other kind of damage)
responsible for subsequent
biological effects.
Track structure model
Primary energy transfer events (excitation
and ionization) occur within 10-10s after
radiation passes through a medium(cell).
Deposition of energy is a stochastic
(quantum mechanical) process, and its spatial
distribution is called track structure.
It is based on the event by event Mote Carlo
method to give the spatial distribution of the
energy deposit in the nanometer scale and
evolution with time in the scale of pico second.
Track structure model is one of the
most important theoretical tools for
studying radiation damage.
Track structure is nano-meter scale
dosimetry can provide the spatial
distribution of the energy deposit in the
scale of nano-meter (dose distribution)
which is the basis for studying the DNA
damage and also for making plan of
cancer radiation therapy
. So far, track structure model is still
incomplete, within this model medium
is mostly treated as water vapor, and
the cutoff energy of secondary electron
is >10eV
Nuclear process such as
fragmentation might be important
More and better cross section data
are always needed.
From Track Structure to Cell Survival
• The DNA damage spectra produced by
ionizing radiation is often obtained
from track structure in water combined
with geometrical models of the DNA.
• These spectra can be provided as the
input data for the phenomenological
dynamical models to bridge the
physical stage and the biological
stage.
Biological Network
• Biochemical networks
Example :P53-Mdm2 interaction
• Cell is best described as a complex
network of chemicals (proteins) connected
by chemical reactions.
• A network consists of nodes connected by
links.
The networks have complex topology. Biological
networks share common topological features with
non-biological networks (Small world, scale
free,etc.).
Network Motifs
One of the major goals in systems
biology is to understand complex protein
networks within living cells. Great
simplification would occur if networks
could be broken down into basic building
block, such as recently defined ‘network
motifs’.
DNA Damage Network & p53-Mdm2
Interaction-A good example how to treat a
complex system in a simple way (network motifs).
P53 Tumor Supressor
Input Signal
+
The Important Functions of P53
• The P53 tumor-suppressor gene
integrates numerous signals that
control cell life and death.
• P53 genes act as brakes to the cycle
of cell growth, DNA replication and
division into two new cells.
激活P53的上游事件
１离子辐射造成的
DNA 损伤激活．
ATM,Chk2参与
２化学药物，紫外
线，蛋白激酶抑制
剂诱导
ATR ,Casein参与
３异常细胞生长
p14ARF参与
三条途径最终都会抑制蛋白的降解，使P53维持高水平，P53通过
调节周围基因的表达，行使抑制细胞分裂和促进细胞调亡的功能．
39
P53 dynamics
• P53 acts as a node for integrating the
incoming damage signal and its primary
function is to act as a transcription factor.
• The biological end-points of P53 induction
are growth arrest or apoptosis.
Previous works on the p53-MDM2 interaction that
gives damped oscillations in a population of cells
a, Generation of oscillations by the p53-Mdm2
feedback loop :A theoretical and experimental
study Ruth Lev Bar-Or et al,PNAS,97(2000)11250
Experimental Results by Ruth Lev Bar-Or
Theoretical results by Ruth Lev Bar-Or
damped oscillation
Lahav et al set out to address what was
happening in individual cells using fluorescently
labeled versions of p53 and Mdm2 that can be
visualized and quantified (2004).
They found that, in response to ionizing
radiation, p53 was expressed a series of discrete
pulses.Genetically identical cells had different
numbers of pulses: zero,one,two or more.
The mean height and duration of each pulse
were fixed and did not depend on the amount of
DNA damage.
The mean number of pulses, however, increased
with DNA damage.
The suggests that the p53-Mdm2 feedback loop
generates a ‘digital’ clock that releases well-timed
quanta of p53 until damage is repaired or the cell dies.
Different cells from the same clone in the
same field of view showed different numbers
of oscillations (Figs. a-c).Figs. d-e shows
a cell with two pulses, p53-CFP first
appeared in the nucleus, followed, after a
delay, by Mdm2-YFP.
p53-MDM2 dynamics in single cells
under stress
MCF-7 cells,
Lahav et al,
Nat. Genet. 2004
The cells tend to show more pulses as damage increases. The mean
height and width of each pulse was constant and did not depend on
irradiation dose.
“Analog”
“Digital”
p53-Mdm2 负反馈环
suppress
实验观测：
promote
在DNA受到损伤时，p53以一
系列脉冲进行表达。每一个峰
的高度和宽度大致是固定的，
不依赖于DNA的受损程度。但
峰的平均数量却依赖于DNA的
受损程度。
确定性模型
考虑单个细胞情形：
dP (t )
 S p   p M (t ) P (t )(1   p S (t ))   p P (t )
dt
dM (t )
 S M   M (t )   M M (t )
dt
P (t   ) N
(t )  N
山形函数 (延迟负反馈)
K  P (t   ) N
信号函数
1(ift  n th )
S (t )  (t  n th )  
0(otherwise)
Shiwei Yan, Yizhong Zhuo. Physica D, 2006, 220(2): 157-162.
New Experiments (fluctuation)(2006)
• Some cells, damaged by either 5 or 10Gy, show sustained oscillations for
the entire time of observation (up to 10 peaks in 60 h). Peak amplitude is
highly variable, whereas peak timing is more precise.
• Radiation dose determines not the number of pulses but rather the
probability that an irradiated cell oscillates permanently or not.
• Whether the damage is large, small or nonexistent, some cells show
“irregular” fluctuations of p53 and Mdm2 with a broad distribution of
interpulse intervals from 8–12 h. (Some cells show non-oscillatory
fluctuations)
随机性模型
d Pr(nP , nM , t )
  (nP , nM ) Pr( nP , nM , t )
dt


mPN
1
 M   N
E

M  1 Pr( nP , nM , t ; mP , mM , t   ),
N
 mP
mP  0 mM  0 K
其中
 (nP , nM )  S P  EP1  1   P nM 1   P S (t )    P   EP  1 nP
 S M  EM1  1   M  EM  1 nM .
The different dynamics of P53 in individual
cells in response to UV with a single pulse
that increases in amplitude and duration in
proportion to the UV dose are observed.
This response contrasts with the previous
described series of fixed pulse in response
to -radiation.
It is also found that while -triggered P53
pulses are excitable, the P53 response to
UV is not excitable and depends on
continuous signaling from the input-sensing
kinaises.
Newest experiment on the response of of P53 to UV radiation (2011)
• Discussions
• a, The behaviors of the oscillations is different
for a population of cells (damped)and individual
cell (sustained) can be described in a unified
way by the proposed dephasing consideration.
• This consideration may be too simple!
(synchronization and coupling?)
• b, Following the “bottom-up approach” P53Mdm2 feedback loop is only a small part of the
DNA damage network , how to take account of
the rest part of the network step by step and
what is the dynamical behavior of the additional
components in the p53 dynamical network.?
c, The different dynamical behaviors due to
the different kinds of stimulation and
different kinds of target cell ,just like the
problem of the “combination of the
projectiles and targets” in nuclear physics
are needed to be further explored both
experimentally and theoretically.
d, It is still far away from full understanding
of the radiation response at cell level, even
at molecular level! A recent report that
mutated p53 protein can gain new function
and also behave as an oncogene.
(B) 主动布朗运动(Active Brownian motion)
Molecular Motor
Surprised at the mechanism how the
chemical energy is converted into directed
motion (mechanical energy) with such high
efficiency.
It became a hot topic in last 90s.
Dozens of different motor proteins coexist
in every eucaryotic cell.
They differ in the type of filament they bind
to(either actin or microtubules), the
direction they move along the filament, and
the “cargo” they carry.
They carry membrane-enclosed
organeles-- such as mitochondria , Golgi
stacks, or secretory vesicles to their
appropriate locations in the cell.
All kind of these remarkable motor
proteins bind to a polarized cytoskeletal
filament and use the energy derived from
repeated cycles of ATP hydrolysis to
move steadily along it through a largescale conformational change in a motor
protein.
Through a mechanical cycle consisting
a few state, the motor protein and its
associated cargo move one step at a
time along the filament (~ typically a few
nanometers).
ATP—adenosine triphosphate
(三磷酸腺苷)
Figure 3-77 Molecular Biology of the Cell (© Garland Science 2008)
(1), Introduction:
Typical example of development along
Tech.---------Exp.--------------theories
Last decade :
Optical tweezers, Video microscopy etc.
Single molecular assays: (resol.nm, ms)
Structure (conformational changes),
Kinetics: V~ATP, V~Loads (low),
Energetic: Efficiencies
Theoretical models:Brownian motors
~Brownian rectifier
Physical Considerations
(1)Small size ~ 100Å (10nm),
(2)Large Brownian Motion,
(3)Binding energy ~ kT,
(4)Nonequilibrium at each Protein site
ATP →ADP +Pi
(5)Filaments are periodic and rigid polar structures
Small machines with large noise
Brownian motors consist of three basic
ingredients:
asymmetric periodic potential
(ratchet potential)
stochastic forces due to thermal
fluctuations
various non-equilibrium sources,
which lead to various models
Most Successful Models:
(Mainly on kinetics)
1, Flashing ratchet model
( More mechanical )
2, Master equation approaches
(More chemical)
They can explain experimental data
well!
3, Their relationship is needed to be
further explored!
(2), Discussions and Perspectives
New challenge and opportunity!
1,To apply the ratcheting mechanism to
various problems, such as translocation
of proteins across membranes through a
nanopore.
2,To describe the behavior of groups of
molecular motors, in particular the
mechanism of coordination between
molecular motors,as molecular motors do
not work in isolation but in groups in vivo.
In muscles, the number of myosins can
19
10
reach a few
,even in intracellular transport,
about 10 motors coordinate their motion in
order to transport vesicles.
“More is different.”
Theoretical models have shown that
interaction between motors can lead to
nontrivial collective behaviors which cannot
be explained by only considering the
behavior of an individual motor. These
effects include bidirectional motion,
oscillations, hysteresis and the formation of
dynamical patterns.
However, the collective effects due to
dynamical phase emerge naturally in the
limit of an infinite number of motors for
which fluctuations average out.
More precise theories considering a finite
number of motors are clearly needed to
bridge the gap between single molecule
descriptions and the limit of large motor
collections.
α = 1.5 (superdiffusion)
Phys.Rev.Lett.85,5655(2000)
Sub-diffusive motion of RNA molecules in the cell.
Phys.Rev.Lett.96,098102(2006)
(3), Collective cell migration
A number of biological processes, such
as embryo development, cancer
metastasis or wound healing, rely on cells
moving in concert. The mechanisms
leading to the emergence of coordinated
motion remain however largely
unexplored.
C,Perspective:
Biological problems are full of mysteries.
Despite all of progress has been made so
far, the question “What is life” remains.
In a living system we need such a
theory :given the conditions of a living
system at t1 , one would be able to make a
quantitative prediction about that system at
t2 , when it runs out of something or other.
Equations of Life!
According Feynman’s idea of
understanding: “ That which I cannot create I
do not understand”.
Learning by building!
We are much better at taking cells apart(top
down) than putting them together (bottom
up).
Reconstitution of biological processes from
component molecules has been a powerful
but difficult approach to studying functional
organization in biology.
The coupling of the membrane (the sensory
organ) and the cytoskeleton ( which
maintains stability) is crucial for survival of
a cell.
In terms of the bottom-up approach, this
means that active networks need to be
confined within vesicles and coupled to their
composite membranes in a dynamic manner.
Such an achievement performed in the
laboratory would result in the first “artificial
cell”
Artificial cells – a new horizon?
The artificial cell is an emerging problem,
the solution of which will only arise from a
concerted effort of the whole natural
sciences community.
Physics certainly will play its important
role as before.
“Biology is a rapidly becoming a science
that demands more intense mathematical
and physical analysis than biologists have
been accustomed to, and such analysis will
be required to understand the workings of
cells.”
Self-assembled, passive actin networks have
been reconstituted within a phospholipid vesicle.
谢谢诸位!
2, New approaches to the solutions of
noisy biological networks
(A) How to look at the biological networks
An important goal in the post-genomic
research is to uncover the fundamental
design principles that provide the common
underlying structure and function in all cell
and microorganisms.
(a) Complex network is responsible for
the biological functions in cells.
The interactions among all molecules (such
as proteins, DNA, RNA and small molecules)
with their very noisy environment, within a
living cell form a complex network.
As most genes and proteins do not have a
function on their own, rather they acquire a
specific role through the complex network
of interaction with other proteins and genes.
(b) Division of the total system into system
of interest and environment
Generally, a complex molecular network is
intertwined among various processes,
including gene regulation, protein
interactions ,metabolism, and signal
transduction. Each process is itself a
network within the complex network
‘network of networks’.
Interactions in each process (or a specific
network) are relatively strongly correlated ,
but interactions between these processes
are relatively weak and independent.
Therefore we only need to couple all the
reactions involved in process of interest,
i.e. to divide the total system into a system
of interest and the environment.
(c) Intrinsic and extrinsic noises
Noise arising from the inherently
probabilistic reactions within a system is
typically called intrinsic or internal,
whereas the effect of environmental
fluctuations on a system is called extrinsic
or external.
(B) Theoretical tools-------Stochastic
representation
A general elementary biochemical
reactions can be represented as follows
r1R1 +r2R2 + ∙ ∙ + rmRm→ p1P2+p2P2+ ∙ ∙ ∙ +
pnPn
It is possible to write down a master
equations for a general molecular network.
For a concrete problem:
Step1, Write down the basic biochemical
equations.
Step2, Draw the graphical representation.
Step3, Write down the corresponding
Master Equations.
Step4, Find the solutions to the Master
Equations.
To find a new way to the solutions!
Example:
A variational approach
J. Chem.Phys.125,p.123106(2006)
The simplest enzymatic signal amplification
process
This is a binary system .To write the master
equation, we denote P(m,n) the probability
of having m R*’s and n A’s,then the Master
equations can be expressed as
Where N = A + A*.
This simple two-step cascade in
commonly found embedded in the onset of
a reaction pathway of many important
signaling cascades.
This master equation actually contains
infinite coupled ODEs.
The QFT formulation
Other states are built up from the vacuum
state, such as the n-particle state |n.
Hence,
is the particle number operator.
Notice that the state |n are not normalized
in the usual sense, since
but they are orthogonal
The state that corresponds to a probability
distribution P(n) is
The probabilities are thus encoded into
the coefficients of different particle
number states superimposed into the
“wave function” |.
In order to compute the physical
observables, the harvesting state
Is introduced. It is easy to check that
They correspond to normalization,
probability conservation
And the mth moment of a particle
number, respectively.
The evolution of probabilities is
governed by a wave equation for |:
Where b,b are creation and annihilation
operators associated with R and a,a with
A. In this case,
In contrast to ordinary quantum mechanics,
the operator  is non- Hermitian, so the
inner products between the states are not
conserved. This was the reason to introduce
the harvesting state.
Nevertheless, many QFT techniques still
can be profitably applied albeit with some
modification.
(C) Perspectives
(a) It seems worth to explore to apply the
projection method to the noisy system
described by Master equations.
(b)The new ingredients are :when the
total system is divided into system of
interest and the environment, the intrinsic
noise is there and the extrinsic noise
depends both on the coupling as well as
the stochastic nature of the environment.
(c ) 1, General formulation
2, Solvable models
3, Concrete problems