Changes to the advertised
schedule
In the Course Guide ...
Topic 1 Energetics and Protein Function
Topic 2 Protein Movement
Topic 3 Protein Purification
Will Actually Be Delivered ...
Topic 1 Introduction to the In Vitro Study of Proteins
Topic 2 Energetics and Protein Function
Topic 3 Protein Purification
Revised topic outlines will be posted to Cecil
Monday, 2 March 15
1
INTRODUCTION TO THE IN
VITRO STUDY OF PROTEINS
BIOSCI 350 Topic 1.
Monday, 2 March 15
2
Proteins don’t
know biology
Biology is largely driven by the
amazing tasks that proteins perform
But proteins don’t know biology !!
The influential American protein
chemist Charles Tanford termed
proteins “Nature’s Robots”
From Ernst Haeckel. Kunstformen der Natur (Art forms of Nature). 1904.
Monday, 2 March 15
3
Nature’s
Robots ?
Robots are automatons or selfoperating machines ... once
programmed you don’t need to tell
them what to do, they already know.
Proteins are the self-operating
machines which carry out biology.
They are “programmed” by their
amino acid composition.
Sculpture by Mike Rivamonte: http://www.rivamonterobots.com
Monday, 2 March 15
4
This reductive approach is easy to
mock ...
http://xkcd.com
However proteins are simply large molecules which operate according to the
laws of chemistry and physics. This is a basic precept of Molecular Biology.
And this concept underpins what we teach you in BIOSCI 350.
Reductive analysis of proteins may not explain all of biology for you. But it will
tell you what is physically possible
It tells you what nature’s robots can do.
Monday, 2 March 15
5
Jiggling and
Wiggling
One of the primary things proteins do is fluctuate in
conformation Even when “at rest” proteins, are highly
dynamic entities.
As Richard Feynman (Famous Physicist) once noted
“…Everything that living things do can be understood in
terms of the jiggling and wiggling of atoms… ”
When energy is supplied in a useable form (e.g as ATP)
proteins can do remarkable things, such as undertake
directed motion (directional and coordinated jiggling and
wiggling)
To illustrate directed motion by proteins, as well as their
“robotic” nature, we’ll look at some remarkable results for
the motor proteins that transport cargo along the actin
cytoskeleton.
Monday, 2 March 15
Sculpture by Mike Rivamonte: http://www.rivamonterobots.com
6
Actin and
Myosin
Actin filaments - in association with
the molecular motor Myosin - are
responsible for muscle contraction.
The actin cytoskeleton is also
central to the organization and
motility of non muscle cells.
Actin filaments visualized in the cell using super resolution
fluorescence microscopy. Xu, K., Babcock, H.P., and Zhuang, X.
(2012). Dual-objective STORM reveals three-dimensional filament
organization in the actin cytoskeleton. Nat Meth 9, 185–188.
Monday, 2 March 15
7
Myosin
Specialist myosins (e.g Myosins V and
VI) carry cargo along actin filaments
within a cell.
The cargo carrying Myosins V and VI)
are “two-legged” and “step” along actin
filaments in a processive manner
analogous to human walking.
All myosins are ATP-dependent
molecular motors. Directed movement
requires the consumption of stored
chemical energy.
Monday, 2 March 15
Fig
sin
(A)
a
Ca
ex
tha
by
by
dim
et
by
thi
tai
co
res
tai
(B
et
as
stu
nu
of myosin VI immediately following the lever arm is largely Finally, we performed
a helical and forms a highly compacted domain, which the quenching and deletion
authors of
postulated
was most
consistent
with a three-helix
evidence that t
Model
Full-Length
Myosin
VI bound
to an actindefinitive
filament.
bundle. They further suggested that the region in between the
Mukherjea,
M. et al. Myosin VI Dimerization Triggers an Unfolding of a
three-helix bundle and the cargo-binding domain is a stable RESULTS
SAH that Bundle
forms thein
bulk
of thetoLAE
necessary
for the large
Three-Helix
Order
Extend
Its Reach.
Mol Cell (2009).doi:
step sizes of myosin VI. Lastly, they postulated that the cargo- The Proximal Tail Is a
10.1016/j.molcel.2009.07.010.
binding domain of full-length myosin VI is solely responsible for Crystallization of a myos
dimerization (see Figure 1). However, this postulate is inconsis- pressed with CaM gave r
tent with our earlier observations (Park et al., 2006) that some- tained only residues 770
where proximal to the cargo-binding domain is a region that on data collection and re
can allow dimerization of myosin VI, resulting in processive evident that we crystalliz
step sizes identical to the full-length dimer. The data of Park nally expressed construc
et al. (2006) might be compatible with a model in which a SAH lowed the determination
domain provides the LAE of myosin VI if there is a residual (FLA) containing two CaM
8
Atomic Force
microscopy
In this technique the specimen is
scanned with a mechanical probe.
The interaction between probe and
surface is used to derive an image of
the surface.
Figure from Nölting, Methods in Modern Biophysics
Some beautiful results for membrane proteins ...
Müller, D.J., and Dufrêne,
Y.F. (2008). Atomic force
microscopy as a
multifunctional molecular
toolbox in
nanobiotechnology. Nature
Nanotech 3, 261–269.
Light is
“high” &
dark is
“low”
Human communication channels
known as gap junctions form
hexameric pores.
Monday, 2 March 15
Bovine rhodopsin, the visual Assembly of light-harvesting ii (small
pigment of the eye, assembles doughnuts) and light-harvesting i
complexes (large doughnuts,
into rows of dimers
surrounding the reaction centre)
9
AFM used to image
Myosin walking along
actin
Myosin is programmed to walk
directionally along actin filaments.
Removed from the cell and fueled with
ATP it carries out that program.
In 2010 in a remarkable experimental
tour de force, Toshio Ando and
colleagues directly visualized single
molecules of Myosin walking along actin
using atomic force microscopy (AFM)
Video Imaging of Myosin walking along Actin filaments. Kodera, N.,
Yamamoto, D., Ishikawa, R., and Ando, T. (2010). Video imaging of
walking myosin V by high-speed atomic force microscopy. Nature 468,
72–76.
Monday, 2 March 15
10
In vitro analysis of
protein function
Hopefully that helps convince you of this:
Because proteins are just large molecules and
their functions are programmed by the
chemistry and physics of their amino acid
chains - you can learn a lot by removing them
from their biological environment and
studying them in vitro using a variety of
biophysical and biochemical techniques
A large part of BIOSCI350 will therefore be
concerned with in vitro protein analysis
Sculpture by Mike Rivamonte: http://www.rivamonterobots.com
Monday, 2 March 15
11
In vitro analysis of protein
function
Proteins Like to Be Wet
Ensemble and Single Molecule Measurements on Proteins
Chemical Equilibrium and The Utility of Equilibrium Studies
The implications of the Really Big Numbers involved in Ensemble
measurements on proteins.
Monday, 2 March 15
12
Proteins like to
be wet
Water is essential for maintaing
the integrity of protein structure.
The only significant exceptions are
fibrous proteins like keratin, that
forms skin, hair and nails.
It follows that proteins are generally
studied in aqueous solutions.
This includes integral membrane
proteins, which are typically reconstituted
into artificial membrane-like
environments to keep them happy.
Structural biologists coax proteins into
forming ordered solids (crystals) to help
generate high resolution structural
information. However the word “solid”
can be deceiving as protein crystals
contain about 50% solvent, on average !!!
Shrek the sheep - encased in quite a bit of keratin
Monday, 2 March 15
13
Ensemble and Single-Molecule
measurements on proteins
When we make experimental observations on proteins one important distinction is
between:
Single-molecule measurements which reflect the behavior of individual
molecules
Ensemble measurements which reflect the average behavior of a very large
numbers of molecules.
Although we probably want to explain the action of individual protein molecules,
ensemble measurements made on protein solutions are often the best we can do.
Monday, 2 March 15
14
Single Molecule Techniques
I’ve already shown one example of single molecule analysis - the spectacular AFM
experiments from Toshio Ando’s group.
It’s worth working through another example, in order to emphasize the
differences between single molecule and ensemble measurements.
The first single molecule experiments on proteins were the patch clamp
experiments devised by Erwin Neher and Bert Sakaman.
The patch clamp technique made it possible to record the currents of single
ion channels.
Monday, 2 March 15
15
Pur i f i ed Sodi umChannel s i n Li pi d Bi l ayer s
KELLER ET AL.
Single molecule analysis of
voltage gated ion channels
KELLER ET AL.
Pur i f i ed Sodi umChannel s i n Li pi d Bi l ayer s
•Voltage gated ion channels mediate the
inward sodium current, during an action
potential (or nerve impulse)
•The top panel shows the current resulting
from a single sodium channel at varying
applied voltages.
B
B
cl osed
cl osed
25 pS
•The bottom panel shows the detail of the
experiment at an applied voltage of -95 mV
i
open
25 pS
x- 10 Ms- - 1
i
open
cl osed
x- 10
Ms- - 1
cl osed
open
t - - 10 m
ss
Vol t age dependence of sodi umchannel gat i ng. ( A) Asi ngl e t r ans- f aci ng
umchannel
was Hartshorne,
open sodi Keller,
i ncor por at ed i R.P.,
nt o a diTalvenheimo,
phyt anoyl PC biJ.A.,
l ayer fCatterall,
or med acr oss a 70B.U.,
Amaper t ur e bat hed i n 0. 5 MNaCl medi umI ( ci s) and 0 . 2 M NaCl medi umI pl us
and
Montal,
M.r ent
(1986).
Sodium
channels
t - - 10 m
ss1 AW.A.,
MBTX(
t r ans)
. The cur
was r ecor
ded under
vol t age-inclplanar
amp condilipid
t i ons whi l e
t he vol Vol
t aget w
asChannel
changed
i n 10- m
V st eps l ast
1 mi n f gat
r om135(channels
gating
kinetics
ofi ng
purified
sodium
- 55 ngl
mV.e The
FI GURE 1 . bilayers.
age
dependence
of
sodi umchannel
i ng.
A)t o Asi
an
cur r ent r ecor ds wer e f i l t er ed at 1 kHz, conver t ed t o di gi t al f or m at a sampl ti rng
modified
by
batrachotoxin.
Physiol.
sodi umchannel
i z,
ncor
att ed
o aJ.diGen.
phyt
PC88,
l1–23.
ayerS char
f ort m
ed acr
f r equency ofwas
5 kH
andpor
pl ot
ed ati ntr educed
speedanoyl
on a Goul
dbi2200
r ecor
der o
Amaper( G
t oul
ur ed, bat
I nc hed
., Cl evel
H) . aC
( B)l The
upper
i n and,
0. 5 O
MN
medi
umIr ecor
( ci ds) i sand
0put
. 2erM
a com
- di N
giaC
t i lzedmedi
si gnalu
r
ecor
ded
at
an
appl
i
ed
vol
um
e
of
V=
m
V
- 95 dedunder
e condi
t i ons
aspi ncondi
A. Af t16
1 AMBTX( t r ans) . The cur r ent was r ecor
undersam
vol
t agecl am
teri o
FI GURE 1 .
Monday, 2 March 15
KELLER ET AL.
Single molecule analysis of
voltage gated ion channels
Pur i f i ed Sodi umChannel s i n Li pi d Bi l ayer s
KELLER ET AL.
Pur i f i ed Sodi umChannel s i n Li pi d Bi l ayer s
It’s apparent that:
•The channel is undergoing rapid and
seemingly random transitions between
conducting and non conducting states
•Channel opening is strongly voltage
dependent
B
✴At large voltages a channel spends most ofB
cl osed
its time in the closed state (no sodium ions cl osed25 pS
pass through, and hence no current)
i
25 pS
open
✴As the voltage decreases, the channel
spends increasing amounts of time in the
open state.
x- 10 Ms- - 1
i
open
cl osed
x- 10
Ms- - 1
cl osed
open
t - - 10 m
ss
Vol t age dependence of sodi umchannel gat i ng. ( A) Asi ngl e t r ans- f aci ng
umchannel
was Hartshorne,
open sodi Keller,
i ncor por at ed i R.P.,
nt o a diTalvenheimo,
phyt anoyl PC biJ.A.,
l ayer fCatterall,
or med acr oss a 70B.U.,
Amaper t ur e bat hed i n 0. 5 MNaCl medi umI ( ci s) and 0 . 2 M NaCl medi umI pl us
and
Montal,
M.r ent
(1986).
Sodium
channels
t - - 10 m
ss1 AW.A.,
MBTX(
t r ans)
. The cur
was r ecor
ded under
vol t age-inclplanar
amp condilipid
t i ons whi l e
t he vol Vol
t aget w
asChannel
changed
i n 10- m
V st eps l ast
1 mi n f gat
r om135(channels
gating
kinetics
ofi ng
purified
sodium
- 55 ngl
mV.e The
FI GURE 1 . bilayers.
age
dependence
of
sodi umchannel
i ng.
A)t o Asi
an
cur r ent r ecor ds wer e f i l t er ed at 1 kHz, conver t ed t o di gi t al f or m at a sampl ti rng
modified
by
batrachotoxin.
Physiol.
sodi umchannel
i z,
ncor
att ed
o aJ.diGen.
phyt
PC88,
l1–23.
ayerS char
f ort m
ed acr
f r equency ofwas
5 kH
andpor
pl ot
ed ati ntr educed
speedanoyl
on a Goul
dbi2200
r ecor
der o
Amaper( G
t oul
ur ed, bat
I nc hed
., Cl evel
H) . aC
( B)l The
upper
i n and,
0. 5 O
MN
medi
umIr ecor
( ci ds) i sand
0put
. 2erM
a com
- di N
giaC
t i lzedmedi
si gnalu
r
ecor
ded
at
an
appl
i
ed
vol
um
e
of
V=
m
V
- 95 dedunder
e condi
t i ons
aspi ncondi
A. Af t17
1 AMBTX( t r ans) . The cur r ent was r ecor
undersam
vol
t agecl am
teri o
FI GURE 1 .
Monday, 2 March 15
Single molecule analysis of
voltage gated ion channels
B
cl osed
25 pS
Detailed analysis of this data yields a wealth of
information about the channel opening mechanism
a
Bottom view
P2
Selectivity
filter
S5
Central
cavity
Side
view
S6
gure 2 | NavAb pore module. a, Pore-lining S6 helices of NavAb (yellow)
Monday, 2 March 15
ARTICLE RESEARCH
Ms- - 1
cl osed
open
NavAb
Kv1.2
c Side
view
S6
S4–S5
linker
t - - 10 m
ss
Vol t age dependence of sodi umchannel gat i ng. ( A) Asi ngl e t r ans- f aci ng
sodi Keller,
umchannel
was Hartshorne,
B.U.,
i ncor por at ed i R.P.,
nt o a diTalvenheimo,
phyt anoyl PC biJ.A.,
l ayer fCatterall,
or med acr oss a 70S65 interaction
site
Amaper t ur e bat hed i n 0.
aCl medi
umI ( cichannels
s) and 0 . 2 in
Mplanar
NaCl medi
umI pl us
W.A., and Montal, M.MN
(1986).
Sodium
lipid
1 AMBTX( t r ans) . The cur r ent was rS4–S5
ecor ded under vol t age- cl amp condi t i ons whi l e
kinetics
ofi ng
purified
t he bilayers.
vol t age wasChannel
changed gating
i n 10- m
V st eps
linker
l ast
1 mi n f sodium
r om- 135channels
t o - 55 mV. The
cur r ent modified
r ecor ds werby
e batrachotoxin.
f i l t er ed at S6
1 kHz,J. conver
ed t o di gi88,
t al f1–23.
or m at a sampl i ng
Gen. tPhysiol.
f r equency of 5 kHz, and pl ot t ed at r educed speed on a Goul d 2200 S char t r ecor der
( Goul d, I nc ., Cl evel and, OH) . ( B) The upper r ecor d i s a comput er - di gi t i zed si gnal
r ecor ded at an appl i ed vol ume of V= - 95 mV under same condi t i ons as i n A. Af t er
f i l t er i ng at 2 kHz, t he rS6
ecorinteraction
ds wer e di site
gi t i zed at a sampl i ng i nt er val of 100 As . A
dow
e nwar d def l ect i on i s a channel openi ng event and t he next upwar d st ep i s
associ at ed wi t h channel cl osi ng. Tr ansi t i ons bet ween t he cl osed and open st at es ar e
Extracellular i ndi cat ed by t he ar r ows . The l ower r ecor d i s t he r econst r uct i on of t he si gnal by a
funnel
pat t er n- r ecogni t i on comput er pr ogr am( Labar ca et al ., 1984) .
P
Pore module of
the voltage
gated Sodium
channel
x- 10
b
The first crystal structure of a voltage gated sodium
channel was solved in 2011 and elegantly explains
NavAb
these observations.
The “activation gate” at the
MlotiK
bottom of the KcsA
central channel is connected
to voltage
S6
NaK
sensing domains (not shown in the figure) which
force the channel open under suitable applied
voltage.
d
i
open
Activation
gate
FI GURE 1 .
Payandeh, J., Scheuer, T.,
Zheng, N., and Catterall,
W.A. (2011). The crystal
structure of a voltage-gated
sodium channel. Nature
475, 353–359.
atoms of Met 221 in NavAb. c, Site for interaction of S6 with S4–S5 linkers (top,
18
convenient to work with the fraction of channels (denoted p) than w
concentration of channels (they differ only by a constant, namely th
concentration of channels in the system). The fraction of channels in a given
also referred to as the occupancy of that state.
Ensemble analysis of
Observed rate constants
voltage gated ion channels Notice that the observed or macroscopic time constant (τ), or its recipro
Of course its also possible to make ensemble measurements
on voltage gated ion channels
For example the figure on the right shows the decay of a
miniature endplate current (MEPC) at the neuromuscular
junction.
Several thousand voltage-gated channels are involved, a large
enough number to produce a smooth curve in which the
contribution of individual channels cannot be detected.
We’ll skip the experimental details. Suffice to say in this
experiment the observed current should be proportional
Fig. 1. Endplate current evoked by nerve stimulation (−130 mV, inward current sho
The observed current is shown by the filled circles; the continuous line is a fi
to the fraction of the channels that are open. downward).
Endplate
evoked
by Reproduced
nerve stimulation
single exponential
curve withcurrent
a time constant
of 7.1 ms.
with permission f
Colquhoun & Sheridan
(1981).
(-130
mV, inward current shown downward).
While there is still lots of information here about channel
activity (and biological function !!) the experiments shed
little light on single channel behavior (The random behavior
of the individual channels is hidden)
Monday, 2 March 15
From Colquhoun and Hawkes. The Interpretation of single channel recordings. In
Microelectrode Techniques: The Plymouth Workshop Handbook 2nd Edition. 1994.
19
Back to ensemble
measurements
Some simple bulk solution properties
Density
Among the simplest ensemble
measurements we can make are
measurements of bulk solution
properties:
Viscosity
Heat Capacity (The amount of heat energy
required to change a solution’s
temperature)
Freezing point ...
Monday, 2 March 15
20
Reasons we might care to
measure such stuff, Part 1
Measuring these simple properties might be
needed to interpret data generated by more
advanced biophysical techniques
e.g. Interpretation of protein sedimentation
experiments in the analytical ultracentrifuge
requires measurement of solution densities
The experiment
involves
spinning your
protein solution
at very high
speed:
Monday, 2 March 15
http://www.uslims.uthscsa.edu
... causing a protein molecule to
experience the following forces:
Buoyant Force
Frictional Force
The sedimentation
force and the
frictional force
depend on protein
shape and mass!!
Sedimentation Force
Density matters
because it
determines the
buoyant force:
21
Reasons we might care to
measure such stuff, Part 2
As a protein is added to a solution at
increasingly high concentrations the
protein will start to change the
solution properties. This may tell us
something informative
Viscosity measurements on protein
solutions can yield information about
protein conformation
We will illustrate these points by
taking a more detailed look at
viscosity
Heat capacity measurements on protein
solutions can yield information about
protein stability.
Monday, 2 March 15
22
Viscosity
•A measure of the
resistance to flow.
•It corresponds to the
informal notion of the
“thickness” of a liquid.
•Viscosity is essentially,
“fluid friction”
FROM EMINA MIRIC, UNIVERSITY OF UTAH
Monday, 2 March 15
23
Watch the units
They are not intuitive, and there are a
lot of minor variations:
Viscosity might be reported in Poise (P):
1 Poise (P) = 1 g.cm-1.sec-1
or centiPoise (cP):
1 centiPoise = 10-2 P = 10-2 g.cm-1.sec-1
or Pascal seconds (Pa.sec):
1 Pa.sec = 1 kg.m-1.sec-1 = 10 P
or milli Pascal seconds (mPa.sec):
1 mPa.sec = 10-3 Pa.sec = 10-3 kg.m-1.sec-1 = 1 cP
Monday, 2 March 15
VISCOSITY OF WATER AT 20 °C
~ 1 cP
VISCOSITY OF GLYCEROL AT 20 °C
~ 1400 cP
VISCOSITY OF MAPLE SYRUP AT
20 °C
~ 3200 cP
24
VISCOSITY IS VERY SENSITIVE TO TEMPERATURE
Monday, 2 March 15
25
When would we need to
measure solution viscosity?
For any technique which follows protein transport through
solution, knowledge of solution viscosity is required to
correctly interpret the results.
Monday, 2 March 15
26
e.g.
Techniques which measure the
speed of proteins diffusing freely:
The Initial Situation
• Dynamic light scattering
• Pulse-field gradient NMR ...
Techniques which measure
the movement of proteins
under an applied force:
Some short time (μS) later.
• Sedimentation velocity
experiments in the
ultracentrifuge ...
Monday, 2 March 15
27
As the protein concentration
increases it starts to to
measurably impact solution
viscosity
This can actually tell us something about protein shape ...
Monday, 2 March 15
28
FROM SERDYUK, ZACCAI AND ZACCAI (2007)
When a solution contains
proteins at significant
concentration, they will
alter solution viscosity.
η/ηo
c
h = h o Q1 + k 1 c + k 2 c + g V
h
2
Q
=
+
1
k
1 c + k 2 c + gV
ho
2
The effect is generally
described by a power series
expansion.
Monday, 2 March 15
η Solution viscosity
ηo Solvent viscosity
c Weight concentration of protein
29
Intrinsic Viscosity
The first order parameter k1 of the
power series expansion is given a
special name.
FROM SERDYUK, ZACCAI AND ZACCAI (2007)
η/ηo
It is termed the intrinsic viscosity
- and it is very sensitive to
molecular shape.
Intrinsic viscosity is generally
given the symbol [η] in the
literature.
c
h
2
Q
=
+
1
k
1 c + k 2 c + gV
ho
Note that helpfully - the “intrinsic viscosity” (k1 or [η]) is not actually a
viscosity & is generally expressed in units cm3/g
Monday, 2 March 15
30
FROM SERDYUK, ZACCAI AND ZACCAI (2007)
INTRINSIC
VISCOSITY
VERSUS
TEMPERATURE
FOR
RIBONUCLEASE
WHEN A PROTEIN UNFOLDS THE POLYPETIDE CHAIN BECOMES AN
EXTENDED AND IRREGULAR COIL. IT’S INTRINSIC VISCOSITY INCREASES
MARKEDLY.
T
STUDYING PROTEIN STABILITY USING
VISCOSITY MEASUREMENTS
Monday, 2 March 15
31
Some Even More Useful
Ensemble Measurements ...
A host of more powerful techniques
involve measuring the interaction of
protein solutions with
electromagnetic radiation of varying
kinds.
Tom Brittain will discuss some of
these
Monday, 2 March 15
Absorption
Scattering
Fluorescence
Polarization
Magnetic resonance
32
Some Even More Useful
Ensemble Measurements ...
X-ray Crystallography is the highest
resolution imaging technique
available in biology (Chris Squire will
talk about it later in the course).
But it’s useful to note that it too, is an
ensemble averaged measurement.
The image that’s formed is an average
one, resulting from the millions of
protein molecules found in the crystal.
This can complicate interpretation:
Monday, 2 March 15
Electron density isosurface associated with an
Arginine side chain that can exist in one of
two conformational states in a protein crystal
Wlodawer et al. Protein crystallography for non-crystallographers, or how to get the best
(but not more) from published macromolecular structures. FEBS J (2007)
33
Chemical
Equilibrium
A lot of experimental investigations
of proteins are done under
equilibrium conditions. But what is
chemical equilibrium? And how
could it be relevant to biology?
http://www.chemheritage.org
Monday, 2 March 15
34
No net change with
time
Chemical Equilibrium is best viewed as state of
balance. In a system at equilibrium there are no
net changes in composition. Every chemical
process is balanced by its inverse.
Protein-ligand binding example:
At equilibrium the concentrations of the
protein [A] the free ligand [B] and the
complex [A.B] do not change with time.
The word net is key. Chemical equilibrium is is
essentially a statistical concept, that emerges
when we average over the behavior of many many
molecules. When a solution is at equilibrium, a
great deal is still happening at the molecular level.
An individual protein molecule will continue
to bind and release ligand at equilibrium.
However there are no changes to be seen at the
macroscopic level. All bulk solution properties
would be constant with time.
However the net rates of complex formation
and dissociation are balanced, so there is no
macroscopic change in composition.
Monday, 2 March 15
However this does not mean changes are not
occurring at the microscopic level.
35
A little math to make things clearer?
Rate constants and Rate equations are used to describe
ensemble kinetic behavior (At a single molecule level the
rate constants kon and koff will reflect the mean lifetimes
of the various species - i.e how long does the complex
A.B hang around before it dissociates?)
kon
koff
d 6A.B@
dt = k ON 6A@6B@ - k OFF 6A.B@
Rate of Association
Rate of
Dissociation
Invoking rate constants we can write a rate
equation (a differential equation) which
expresses the change in the concentration
of the complex with time:
The Association reaction is second order - the rate is proportional to the concentration of two species
The Dissociation reaction is first order - the rate is proportional to the concentration of one species
Monday, 2 March 15
36
A little math to make things clearer?
d 6A.B@
dt = k ON 6A@6B@ - k OFF 6A.B@
Rate of
Association
-
Rate of
Dissociation
Now at chemical equilibrium
concentrations are unchanging so
d[AB]/dt = 0
k ON ! A$!B$ - k OFF ! A.B$ = 0
k ON ! A$!B$ = k OFF ! A.B$
Rate of
Association
Monday, 2 March 15
=
Rate of
Dissociation
“Every chemical
process is balanced
by its inverse”
37
Some biological processes which
can be usefully treated in terms
of chemical equilibrium
•The processes on the right are often
treated in terms of chemical equilibrium.
usefully
•Because biological systems are demonstrably not
at equilibrium (a cell at equilibrium is dead) many
people have a nagging feeling that equilibrium
models cannot be relevant to biology.
•However there are many problems in biology for
which equilibrium models are not just a good
starting point, but the most insightful way to
understand a phenomenon.
•Equilibrium models are useful in biological
settings when certain processes happen much
faster than others. (i.e. their applicability rests on
the separation of time scales)
Monday, 2 March 15
Phillips, Kondev &Theriot. Physical Biology of The Cell
38
When we make ensemble
measurements on protein
solutions, REALLY big numbers
of molecules are involved
The numbers involved in ensemble average
measurements on protein are difficult to
comprehend.
For simplicity, lets say we are making a
measurement on 1ml of protein solution
with the protein concentration at 1 μM (=
1x10-6 mol/L)
There’s about 6 x 1014 protein molecules
in that solution or 600 000 000 000 000
in long hand.
Monday, 2 March 15
"Space," it says, "is big. Really big. You just
won't believe how vastly, hugely,
mindbogglingly big it is. I mean, you may
think it's a long way down the road to the
chemist's, but that's just peanuts to space
39
The implications
of large numbers
The Fine Print
It’s really important that you understand the implications of these huge
numbers. Unfortunately that means we need to do some statistics (but only
the elementary kind)
The Short Story
A two state ion channel, with each
state equally probable, can be
modeled with a coin toss.
Let’s return to our voltage gated ion channel and suppose that it’s strictly
two state (either open or shut)
Then imagine we have a bunch of channels and tune the applied voltage so
that each channel spends 50% of its time open and 50% shut.
Mathematically the problem of describing the channel state can be reduced
to a coin toss.
For any given channel at a fixed point in time, heads says it’s open, and tails
says it’s shut !
Our model is stochastic as it must be (While we can measure the state of
single ion channel, we certainly can’t predict it)
Monday, 2 March 15
40
4 channels and four coins
Channel 1 Channel 2 Channel 3 Channel 4
Let’s start out with a small
collection of channels - say 4.
=
If we examine each channel in
turn, what’s the probability of
any sequence of outcomes?
Each specific sequence is as
likely as any other
=
=
=
Monday, 2 March 15
41
Onward to
Multiplicity
Channel 1 Channel 2 Channel 3 Channel 4
But actually - we don’t really care about the
probability of a particular configuration of our 4
channels (we’ll call this the micro-state)
What we care about is an aggregate or
macroscopic property - how many channels are
open, and how many are shut (will call this the
macro-state)
That’s what’s going to determine the ion current
in a biological situation !!
This is all the possible micro-states with 1
channel shut (these would all give rise to the
same ion current, and hence for our purposes,
they are equivalent)
In other words if we want a macro-state with
one channel shut, there’s 4 micro-states that
achieve this
The macro-state therefore has a multiplicity
of 4
Monday, 2 March 15
42
More micro-states
and macro-states
Now let’s complete the picture.
This system has 16 micro-states and 5 macro-states
We can compute the probability of each Macro-state
(0 channels shut, 1 channel shut, 2 channels shut ...)
quite simply:
P(Macro-state) =
Multiplicity(Macro-state) / Total number of microstates
Monday, 2 March 15
43
The final result, part 1
Four 4 channels here’s the
multiplicity of each of the
macro-states plotted on a graph
If you had a lot of time and
patience you could repeat this
exercise for increasing numbers
of channels
Alternatively you could derive an
equation (the bionomial
distribution) which would give
you the result directly !!
Monday, 2 March 15
44
The final result, part 2
Let’s forget the equation and just plot the result
Notice how each distribution is always centered
on the point with 1/2 heads and 1/2 tails. This is
the point of maximum multiplicity
Notice also that as N becomes large the
distribution becomes extremely sharp.
The multiplicity of the systems with large N drops
precipitously as we move just a little bit away from
the most probable point (50% heads, 50% tails)
We will not prove it, but as N becomes large our
(Binomial) distribution becomes
indistinguishable from a Gaussian (Normal)
distribution
Monday, 2 March 15
N.B if we were to normalize by the total number
of micro-states these graphs of multiplicity
would turn into probability distributions
45
The take home message
While the ion channels are behaving stochastically (randomly) and
we cannot predict their individual behavior, when we have a huge
population of such channels, the predictions we can make about the
population are effectively deterministic.
As N becomes large - 600 000 000 000 000 for example (!!) the state with maximum multiplicity is the state which is
observed - in this case is a 50/50 mix of heads and tails (or open
and shut channels)
Monday, 2 March 15
46
A couple more points
and we’ll stop
We could replay this fun (!) game with
a loaded dice and nothing would really
change. The system would still tend to
the state of maximum multiplicity
With the probabilities as shown on the
left we would end up with a 70/30 mix
of open and shut channels at large N.
Monday, 2 March 15
X
X
p=0.7
p=0.3
47
And what about
energy?
This probabilistic description is all very well, but can we
relate the probabilities of our two states to their
energies?
Intuition would tell us that if each state of an ion
channel (open or shut) has the same probability of
occurring, it should have the same energy.
And if one state of an ion channel has a higher
probability of occurring, it should have a lower
energy.
Monday, 2 March 15
48
Boltzmann
Thermodynamics of Biological Processes
It turns out that solving that problem is seriously
difficult, but it was done by Ludwig Boltzmann before the
molecular nature of matter was even fully accepted.
A
10 ms
2 pA
C
O
This is one of the greatest acts of scientific genius ever.
B
State
Energy
Weight
Exploring this idea would takes us deep into the world of
statistical mechanics, a step too far for this course
eclosed
e–βeclosed
However we can just inspect the results of Boltzmann’s
great leap and check that they make sense to us.
eopen
e–βeopen
Monday, 2 March 15
Figure
2.2
States
weights
for
ion chann
Garcia,
H.G., Kondev,
J., Orme,and
N., Theriot,
J.A., and Phillips,
R. (2011).
Thermodynamics of biological processes. Meth Enzymol 492, 27–59.
how the
channel transitions back and forth be
et al., 1986). By evaluating the fraction of time
compute the open (and closed) probability. (B
corresponding energies and the Boltzmann w
channel being open as a function of the differen
states. This difference in energy can be tune
49
B
Equation derived from
the Boltzmann treatment
Thermodynamics of Biological Processes
p open = A - bf10 ms
- bf
e
+e
open
closed
2 pA
B
1
=
- bDf
1+e
Closed
Open
where
Energy
Δε State
= εclosed - εopen
(J)
and
eclosed
1
b = k T eopen
B
Weight
e–βeclosed
e–βeopen
Energy
We
eclosed
e–βe
eopen
31
e–β
Figure 2.2 States and weights for ion
C the channel transitions back and f
how
1
et al.,
0.9 1986). By evaluating the fraction o
compute
the open (and closed) probab
0.8
0.7
corresponding
energies and the Boltzm
0.6
channel
being open as a function of the d
0.5
states.
0.4 This difference in energy can
voltages
and tension on the membrane.
0.3
popen
e
- bf open
State
0.2
0.1
idea
that all the channels in a popu
0
−10 −8the
−6 −4
−2 0 2is 4assumed
6 8 10 to
words,
system
∆e = e closed-eopen (kBT )
probability for a single channel ex
the average
fraction of channels in a
Garcia, H.G., Kondev, J., Orme, N., Theriot, J.A., and Phillips, R.
kB = the Boltzmann constant = 1.38 x 10-23 J/K
(2011).
Thermodynamics
of biological
processes.
Meth Enzymol
Figure 2.2 States and weights for ion channel dynamics.
(A)
Current
trace
showing
given
instant.
For
cases
where th
492,
27–59.
how the channel transitions back and forth between reasonable),
the open and statistical
closed states
(Keller te
mechanics
T = By
Temperature
(K)
et al., 1986).
evaluating the
fraction of time spent in either of these two states, we can
the states
from their
compute the open (and closed) probability. (B) Stateseach
of theoftwo-state
ion channel,
the ene
from
their probabilities.
corresponding energies and the Boltzmann weights.energies
(C) Plot of
the probability
of the
to thebetween
same cartoon
channel being open as a function of the difference in energy
the opendepiction
and closedof50th
Monday, 2 March 15
What do you need to know?
After reviewing the material for this topic, and doing some reading and thinking, you
should be able to recognize and explain:
The difference between ensemble and single molecule measurements.
Some reasons we might want to measure simple solution properties, with solution
viscosity as a specific more detailed example.
The nature of the equilibrium state and the reasons equilibrium studies are relevant to
biology.
Why we can predict ensemble or collective behavior of molecules very accurately even
when we can’t predict the state of any individual molecule.
That a mathematical connection can be made between the probability of a molecular state
and its energy (though you don’t have to understand the derivation of this result).
Monday, 2 March 15
51