Overview of next five lectures:
How is directional motility accomplished at the single cell level?
An emphasis on experimental approaches for testing models.
April 14: Bacterial chemotaxis: Description and modeling
Concept of temporal sensing of gradients
Components, General scheme. Mathematical modeling of adaptation and robustness.
April 19: Bacterial chemotaxis: Manipulation & Modeling
Genetic manipulation of the system to test robustness model
Explaining Ultrasensitivity and range of sensitivity.
April 21: D. discodium chemotaxis: Description
cAMP-dependent clustering, formation of fruiting body
Concept of spatial and temporal sensing of gradients
Components
April 26: D. discodium chemotaxis
Genetic manipulation of components
Intracellular gradients of components
Modeling and testing models
April 28: Chemotaxis in neutrophils
Comparison of neutrophils to D. discodium
Components
How do bacteria decide where to go?
QuickTime™ and a
Sorenson Video decompressor
are needed to see this picture.
Salmonella typhimurum
Qu i c k T i m e ™ a n d a
T IF F (U n c o m p r e s s e d ) d e c o m p re s s o r
a re n e e d e d to s e e th i s p i c t u re .
Chemotaxis
toward serine
Chemotaxis away
from phenol
From B.A. Rubik and D.E. Koshland, PNAS 75:2820-2824, 1978
Complex behavior in a uniform
mixture of two chemoattractants
http://www.aip.org.pt/jan00/berg.htm
How do Eukaryotic cells know where to go?
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Video decompressor
are needed to see this picture.
Neutrophil chasing a bacterium:
Stossel, 1999
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None decompre ssor
are need ed to se e th is p icture.
Neutrophil chasing a pipet
Visualization of E. coli flagella with GFP fusion protein
(from Berg lab)
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Cinepak decompressor
are needed to see this picture.
Real time
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Slowed down
<http://www.rowland.harvard.edu/labs/bacteria/index_movies.html
E. Coli anchored via flagella reveals switching
between clockwise and counter-clockwise rotation
(from Berg lab)
QuickTime™ and a
Cinepak decompressor
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<http://www.rowland.harvard.edu/labs/bacteria/index_movies.html
Mysteries of bacterial chemotaxis
1. Bacteria cannot sense direction directly and can only
decide whether to go straight or to stop and randomly chose a
new direction - yet they end up over time at the location of
the highest concentration of chemoattractant. How does this
happen?
Mysteries of bacterial chemotaxis
2) Bacteria can respond to small differences (<1% front to back)
in chemoattractant concentration over a very large (1000
fold) range of basal concentrations. Given what we know
about receptor saturation, how can this work?
Inconsistent with simple hyperbolic
990-1000 or sigmoidal saturation of receptor binding
9.9-10
99-100
A B C D
Receptor
Occupation
E F
AB
EF
CD
[Attractant]
Mysteries of bacterial chemotaxis
3) Bacteria demonstration perfect adaptation: When acutely
given a uniform concentration of a chemotactic agent, they
decrease their tumbling frequency within seconds but after
several minutes return to the exact same tumbling frequency
as before addition of chemotactic agent. A second jump to
yet a higher concentration of chemotactic agent elicits an
idential response.
3B) Adaptation is Robust: Large changes in the
concentrations of the components of the adaptation system
affect the time required to adapt but do not interfere with
perfect adaptation.
Observations
Bacteria move towards attractants
A bacterium senses differences in concentration with time rather than
differences from one end of the bacterium to the other (<1 mm).
Bacteria run (straight line) then tumble (random reorientation).
In a linear gradient of chemoattractant, a 0.2 % difference over 1 mm length
of bacterium becomes a 4% difference over a 20 mm run.
By comparing receptor occupancy at 30 sec to occupancy at time 0, the
bacterium can decide whether it is running in the right direction.
Run = coordinated counterclockwise flagellar rotation
Tumble = uncoordinated clockwise flagellar rotation
Increasing [attractant] -> rarer tumbling -> net movement up gradient.
Other considerations:
Brownian drift (thermal collisions with water, solutes, macromolecules)
alters course from straight line during run. Periodic tumbling ultimately
insures that the steepest path up the chemotactic gradient is ultimately taken.
Bacteria have no significant momentum in a given direction. When flagellar
rotation is switched to clockwise they ‘stop on a dime’.
Adaptation
Alon, U., et al. (1999).
Nature 397: 168-71.
Bacteria rapidly reduce the frequency at which they tumble in
response to elevation of attractant but in the presence of high
constant attractant concentration, they invariably return to the
original tumbling frequency (within 1%). This adaptation
time is variable.
Mutational approaches to tease
out genes mediating chemotaxis
Plate contains serine and aspartate
Wild type
Looking for Asp (2nd choice)
Looking for Ser (1st choice)
Non-motile flg
Ser detection- tsr
Asp detection- tar
http://www.aip.org.pt/jan00/berg.htm
Mutant classes
Lack flagella: flg, flh, fli, flj
Non-motile: mot
General chemotaxis block: che (A, B, R, W, Y, Z)
Specific chemotaxis block: tsr, tar, etc.
Structure of the machinery 1
http://www.aip.org.pt/jan00/berg.htm
Rotary 8 cylinder motor driven by protonmotive force (400 steps/rev)
Structure of the machinery 2
Alberts et al. Mol Biol Cell 3rd Ed.
Structure of the machinery 3
Counter-Clockwise = straight swim,
bundled flagella
Clockwise = tumble, separate flagella
CheY-P binding to motor -> CW rotation
Repellent -> CheA-P -> CheY-P ->  CW
Attractant -> CheA-P -> CheY-P -> CW
Alberts et al. Mol Biol Cell 3rd Ed.
Falk and Hazelbauer (2001) TIBS 26, 257
Two levels of receptor regulation
Attractant binding -> Lower activity
Methylation -> Higher activity
Feedback produces adaptation
Attractant binding reduces the fraction of
active receptors, reduces phosphorylation
of the demethylase, CheB and ultimately
leads to a net increase in receptor
methylation. Since methylated receptors
are active even when attractant is bound,
the activity is ultimately restored to normal
Acute removal of attractant
increases the activity of the
receptors but ultimatley triggers
demethylation and reduces
activity to normal.
Simplified adaptation model with the key assumption that CheB can only
demethylate the active form of the receptor/kinase complex. In this model,
when fewer receptors are active due to an acute increase in ligand binding, CheB
has less substrate available so demethylation is slowed while methylation (CheR)
is constant resulting in a net increase in receptor methylation over the next few
minutes. Since methylation stabilizes the activation state of the receptor (even
when ligand is bound), the net activation ultimately returns to the original value.
This model argues that availability of “active” receptor substrate for CheB is
more important for perfect adaptation than is the phosphorylation state
(activity) of CheB.
Barkai&Leibler 1997 Naure 387, 913
In the absence of ligand, the activity of the system at steady state can be defined by:
Ast = aE + amEm = medium
Where E and Em are the concentrations of unmethylated and methylated receptors, a
is the probability that the unmodified receptor is in the active state and am is the
probability that the methylated receptor is in the active state. In general am is much
higher than a (indicated by size of symbol) so most activity is due to the methylated
receptors.
The activity of the system immediately after addition of saturating ligand (all receptors
bound to ligand) is defined by A(l):
A(l) = a(l)E + am(l)Em = low
Where a(l) is the probability that the unmodified receptor (ligand bound) is in the active
state and am(l) is the probability that the methylated receptor (ligand bound) is in the
active state. In general a(l) and am(l) are smaller than a and am (indicated by size of
symbols) and the ratio of Em to E does not change in the initial seconds after ligand
addition due to the slow rate of methylation/demethylation so ligand binding causes an
acute drop in activity.
However, since CheB can only act on active Em, and less Em is active, demethylation is
reduced while methylation continues, shifting to a higher ratio of Em to E, reaching a new
steady state that has the same activity (but different E/EM ratio) as the original:
A(l)st = a(l)E + am(l)Em = medium = Ast
For simplicity, assume a = 0 (e.g. only the methylated receptor can
be active under any condition) and assume CheR is saturated, then
the rate of change in the methylated form of E is:
dEm/dt = VRmax - VBmax(A)/(Kb+A)
Where Kb is the Michaelis constant for interaction of the activated
form of Em (e.g. A) with CheB. VRmax and VBmax are the maximal
rates of CheR and CheB respectively. VBmax is assumed to be
greater than VRmax.
When dEm/dt = 0 (steady state), then:
Ast = KbVRmax/(VBmax-VRmax) = am(l)Estm
Notice that this equation is only dependent on parameters
reflecting the activities of CheR and CheB and the Michaelis
constant for CheB. Thus, regardless of the perturbation due to
binding of ligand, the system will reset to the same steady state
(e.g. If am(l) is 1/10 of am then Estm will increase 10 fold to
compensate).
Assumptions in Baraki&Leibler adaptation model:
1. Ligand binding is rapid compared to methylation/demethylation
and for simplicity (not necessary for conclusions), ligand
binding affinity is the same for all forms of methylated and
unmethylated forms of the receptor.
2. CheB can only demethylate active receptors and the rate of
demethylation is independent of the methylation state or ligand
binding state of the receptor and independent of the
phosphorylation state of CheB itself.
3. CheR can methylate both active and inactive receptors, whether
or not ligand is bound. For greater simplicity it can also be
assumed that CheR is saturated by substrate and methylates all
receptors at the same rate and this does not affect the general
conclusion.
Assumptions in Baraki&Leibler adaptation model:
Notice that the ability of active receptors to cause
phosphorylation of CheB and inactivate CheB is not considered
in this model (VBmax is treated as a constant). Thus, this
regulation is not required for perfect adaptation. If considered,
this regulation would be predicted to increase the steady state
fraction of active receptors but the system would still exhibit
perfect adaptation.
Ast = KbVRmax/(VBmax-VRmax)
But now, VBmax(t) = kphosA(t) - kdephos
At steady state, VBmax(st) = kphosAst - kdephos
Since the system perfectly adapts, VBmax(st) is a constant
Barkai&Leibler 1997 Naure 387, 913
Testing Robustness
Model has 9 rate constants and 3 enzyme concentrations
In each simulation increase or decrease each parameter
2X randomly
Overall parameter Change = Pn,1/Po,1 x Pn,2/Po,2 x ……..
Do 6000 independent simulations
Examine effects on precision and timing of adaptation
add
remove
Modeling the effect of successive addition and removal
(after 20 min.) of 1, 3, 5 and 7 mM of chemoattractant
Barkai&Leibler 1997 Naure 387, 913
All rate constants in the
model can be changed
randomly by 2 fold with
only a ~15% deviation
from perfect adaptation.
A single rate constant
can be changed by
several orders of
magnitude (holding
other parameters
constant) without
significant deviation
from perfect adaptation.
Multiple methylation is
not necessary for
robustness but improves
adaptation response time
Barkai&Leibler 1997 Naure 387, 913
The time required for adaptation varies widely as the rate constants
for the model are randomly changed.
Barkai&Leibler 1997 Naure 387, 913
The model predicts that increasing the amount of CheR (methylase) from
100 to 300 molecules/cell increases the fraction of active receptors at
steady state (Activity) resulting in a larger fraction of time spent tumbling
and also resulting in a shorter time required to recover following addition
of ligand.
100
300
Consistent with simplified model Ast = KbVRMAX/(VBMAX-VRMAX)
= KbVRMAX/VBMAX for VBMAX>>VRMAX
It can also be shown that t 1/VRMAX (see Alon Chapter exercises)
Thus, substituting for VRMAX, t
Kb/VBMAXAst
Barkai&Leibler 1997 Naure 387, 913
Barkai-Leibler Model is a form of Integral Control
Yi, Huang, Simon&Doyle 2000 PNAS 97, 4649
In the classic integral control model (on right),
applied to bacterial chemotaxis, u is the fraction of
receptors that are not bound to ligand (the external
perturbation) and x is the fraction of receptors that
are not methylated due to demethylation of active
receptors by CheB (the response). y1 is the
resulting activity state of the receptor, which is
linearly related to the fraction of receptors unbound
minus the fraction of receptors demethylated y1=
k(u-x) (e.g. methylated and unbound receptors have
the highest probability of being active). k is a
positive constant relating the total activity of the
receptors to the fraction unoccupied and
methylated. y0 is the steady state level of y1.
At steady state, y1 = y0
KbVRmax/(VBmax-VRmax) (from Barkai&Leibler) a function
only of CheR and CheB binding and turnover numbers - independent of ligand
concentration. y is defined as the difference between the activity at time t (y1) and the
activity at steady state (y0). Thus, at steady state, y = 0. Decreased ligand binding
acutely increases u and elevates y1 to a value above y0, giving a transient positive
value for y.
Barkai-Leibler Model is a form of Integral Control
Yi, Huang, Simon&Doyle 2000 PNAS 97, 4649
At steady state, (y = 0) the rate of methylation and demethylation are equal. If one
assumes for simplicity (as did Barkai&Leibler) that CheR is saturated and unaffected
by ligand binding and that the CheB demethylase only acts on active receptors
(whether or not ligand is bound) then the net rate of demethylation at any
instantaneous time will be directly proportional to y (the transient excess in active
receptors over the steady state value). When y = 0 methylation and demethylation
cancel out.
The fraction of demethylated receptors (x) at any time t is then determined by the
number of receptors in the demethylated state at time zero, x0 (e.g. prior to the
unbinding perturbation) plus the number of receptors that get demethylated during the
interval in which the system was perturbed. This latter term is the integral from the
time at which the perturbation (e.g. ligand unbinding) occurred t=0 to time t of ydt.
So x(t) = x0 +
tydt
Notice that y can0 be + or - depending on whether ligand decreases or increases.
Thus dx/dt = y = k(u-x) - y0
At steady state, dx/dt=y=0 and y1=y0
Notice that since k and y0 are constants, an increase in u (rapid release of ligand) is
ultimately offset by a slow decrease in x so that at steady state k(u-x) = y0.
Assumptions/simplifications in integral model:
1. CheB only acts on active receptors (essential for perfect
adaptation with robustness).
2. The activity of the unmethylated receptor is negligible
compared to methylated.
3. The binding of methylase CheR to receptors is not affected
by ligand binding.
4. The Vmax values of the methylase and demethylase are
independent of receptor occupancy or methylation state.
Variations from these assumptions compromises perfect
adaptation.
The basics of chemotaxis
Receptor:CheW:CheA phosphorylates CheY
Phosphorylated CheY interacts with motor to promote CW rotation and tumble
CheY dephosphorylated by CheZ
Attractant binding reduces CheA activity -> less CheY-P -> rarer tumbling
Repellent binding increases CheA activity -> more CheY-P -> more tumbling
Adaptation via control of methylation:
Ligand binding and receptor methylation jointly control CheA activity
At given ligand occupancy, more methylation -> more CheA activity
At given methylation level, more attractant (less repellent) binding -> less
CheA activity
CheA phosphorylates and activates CheB, the receptor methylase
Attractant -> Less CheA activity -> Less CheB-P -> more methylation ->more
CheA activity
Repellent -> More CheA activity -> More CheB-P -> Less methylation ->Less
CheA activity
Effectively the system measures the difference between the extent of two
processes:
Fast ligand binding
Slow receptor methylation/demethylation
When [attractant] changes fast, the two signals show a large difference &
cells respond