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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 letterIf 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

k1 k2

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 pathwaysA. 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 = a/E1

B = E1*A/E2 = a/E2

C = E2*B/E3 = a/E3

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 pathwaysA. 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!