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What is the point of a signal transduction pathway?
to transmit a signal within a cell, bewteen organelles, or between cells
to amplify, process and integrate information from the extracellular environment to the rest of the cell
To transduce a signal
Why are they so complicated?
Multiple levels of regulation
to allow for regulation and integration of different signals being recieved
Redundancy, crosstalk, regulation of speed and magnitude of response to a stimulus
What I want you to get out of this:
• 1. Signal processing is just as important as signal transduction
• 2. Counterintuitive behavior can arise from simple systems.
• 3. Toy Models (and simulations) can aid your intuition.
• 4. A little bit of math can tell you a lot about a system
Stimulus
mRNA
A simple systemThe hydrogen atom of signaling. Let’s start here with a simple system and then use this to learn approaches and principles.
Stimulus is added at time point zero. Sketch what you think the mRNA abundance over time will look like (your curve should go through the two data points in red).
Steady-state
This is the point whereThe synthesis and degradationRates are matched
Time scale (how long does it take to get “half way” there
Steady-state
This is the point whereThe synthesis and degradationRates are matched
Time scale (how long does it take to get “half way” there
What controls the level of mRNA?
• Synthesis Rate:– Amount of polymerase– Size and length of stimulus– ATP concentration, salt, etc …
• Degradation Rate:– mRNA levels– Nuclease, salts ,etc.
What do we need to follow if we want to “model/understand” the system?
• Only the things that change on the same time scale.
• Side Note:– When of the most important parts of modeling is
it actually makes us think carefully about what we know, what we don’t know, and what we need to measure better to be able to separate between different types of models
Representing our system: simbiology to simulate
mRNAsynthesis degradation
Pictoral representations
3 different synthesis rates
Synthesis rate ONLY effect steady-state
3 different degradation rates
Degradation effects both steady-state and time scale
Even this simple system can have counter intuitive behavior
Correct answer
How to write a differential equation
mRNAsynthesis degradation
Pictoral representations
Equation d(mRNA)dt = synthesis * degradation(mRNA)
This just means: how does the mRNA level change at a given moment in time
Generic Method
A + B C D
One equation for each species (eg each of the letter in your system)
One term for each arrow that points towards or away from a letter
If the arrow points toward it gets a positive sign; if it points away it gets a negative sign
This is multipled by the rate of the arrow (usually written above the arrow)
Finally all the species that are at the BACK side of the arrow are multiplied together(if there are none don’t write anything)
dC/dt = + k1 * A * B - k-1 * C - k2 * C
k1k2
k-1
dD/dt = k2 * C
mRNA sythesis and destruction
d(mRNA)/dt = – a b*mRNA
Steady state
d(mRNA)/dt = – a b*mRNA = 0 mRNA = /a b
Kinetics
mRNA(t) = /a b ( 1 – e- *b t)
Time scale only depends on !b
d(ES)/dt = k1(E)(S) - k-1(ES) – k2(ES)
What happens in a chained chemical reaction: think metabolic pathway
What will happen to the steady-state rate of production of C if we lower the concentration of E2 two-fold?
A. It will increase B. It won't change C. It will decrease D. It depends
Kinetic analysis of molecular pathways
A. Flux conservation in linear pathways
Kinetic analysis of molecular pathways
B. Flux diversion
Finishing enzyme rates
S -> PE
dP/dt = k2 * (ES) d(ES)/dt = k1(E)(S) - k-1(ES) – k2(ES)
Separation of time scales – the quick steps will quickly reach equilibrium
FASTSLOW
Separation of time scales
• What is the distance between me and my friend?
• We both start in San Francisco and go to Boston
• My friend take a plane. I walk.• After 1 day you only really need to know
where I am to know the distance between us.
Finishing enzyme rates
S -> PE
dP/dt = k2 * (ES) d(ES)/dt = k1(E)(S) - k-1(ES) – k2(ES)
FASTSLOW
d(ES)/dt = k1(E)(S) - k-1(ES) – k2(ES) = 0
ES * (k-1+ k2) = k1(E)(S)
ES = k1(E)(S) / (k-1+ k2)
Not that useful because E is an unknown(free enzyme concentration)
Some more math
ET = ES + E
E = ET - ES
ES * (k-1+ k2) = k1(E)(S) = k1(ET - ES)(S)
ES * [(k-1+ k2)+ k1(S)] = k1ET (S)
ES= k1ET (S) / [(k-1+ k2)+ k1(S)] Divide top and bottom by k1
ES = ET (S) / [(k-1+ k2)/k-1+ (S)] Km=(k-1+ k2)/k-1
ES = ET (S) / [Km+ (S)]
ES = ET (S) / [Km+ (S)]
Km>>S ES = ET (S) / Km = a * S
Linear range of enzyme
S>>Km
Saturated enzymeES = ET
Two regimes
In the linear regime
dA = a – E1*A
dB = E1*A – E2*B
dC = E2*B – E3*C
At steady state:Equation
A = / 1a E
B = E1*A/E2 = / 2a E
C = E2*B/E3 = / 3a E
All the concentrations only depend on !a
Wnt signaling is central to stem cell self-renewal but remains poorly understood
bCatbCat
TCF
bCatbCat
bCatbCat
bCatbCatTCF
No Wnt Wnt
bCat
deg.deg.
The core mechanism of b-catenin stabilization by Wnt action is hotly debated
b-cat b-catP
b-catP P P
Pb-cat
P P PP
U U U
DegradationSynthesis
CK1a GSK3 bTrCP
Other mechanisms:• Sequestration of Axin1 (Mao 2001)• Axin1 degradation (Mao 2001, Lee 2003)
• Inhibited (Amit et al., 2002)• Not inhibited (Liu 2002, Li et al. 2012)
• Inhibited (Cselenyi 2008; Piao 2008; Wu et al., 2009; Taelman 2010)
• Not inhibited (Li et al., 2012)
• Inhibited (Li et al., 2012)
Hernandez*, Klein* and Kirschner, Science 2012
Axin/APC
The core mechanism of b-catenin stabilization by Wnt action is hotly debated
b-cat b-catP
b-catP P P
Pb-cat
P P PP
U U U
DegradationSynthesis
CK1a GSK3 bTrCP
Hernandez*, Klein* and Kirschner, Science 2012
Axin/APC
Kinetic analysis of molecular pathways
A. Flux conservation in linear pathways
initial steady state
transient state
new steady
state
Response of b-catenin to Wnt stimulation involves a transition between two steady-states
Hernandez*, Klein* and Kirschner, Science 2012
Kinetic analysis of molecular pathways
B. Flux diversion
What if regulation is upstream and downstream?
Hernandez*, Klein* and Kirschner, Science 2012
Kinetic analysis reveals the points of Wnt action
Hernandez*, Klein* and Kirschner, Science 2012
pT41/S37/S33 b-catenin
b-catenin
Bacterial chemotaxis
• If bacteria sense increasing ligand they swim straight
• If they sense decreasing ligand they turn a random direction.
• Able to chemotax up a gradient of many orders of magnitude. How?
Ligand + receptor isn’t very good
L + R LRk1
k-1
L = ligandR = ReceptorLR – is the complex and active species
Look familiar? This won’t be very responsive
Actual System
That didn’t help
Magic
d(RmL + Rm) = Vm * ( (R + RL)/((R + RL) + Km) – VD * ( (Rm + RmL)/((Rm + RmL) + KD)
d(RmL + Rm) = Vm * (1) – VD * ( (Rm + RmL)/((Rm + RmL) + KD)
Saturated methylase
d(RmL + Rm) = Vm * (1) – VD * ( (RmL)/(RmL + KD) = 0
Demethylase only works on RmL
RmL = Vm * KD/(VD-Vm)
Ligand independent!