dynamic processes in cells(a systems approach to biology)

jeremy gunawardenadepartment of systems biology

harvard medical school

lecture 930 october 2014

state space “landscapes” and parameter “geography”

PARAMETER SPACE

single stable state

two stable and one unstable state

stable limit cycle

STATE SPACES

single stable state

state space landscapes and cellular identity

1905-1975

Conrad Waddington

Slack, “Conrad Hal Waddington: the last renaissance biologist?”, Nature Rev Genetics 3:889-95 2002. Huang, “Reprogramming cell fates: reconciling rarity with robustness”, Bioessays, 31:546-60 2009; Hana, Saha, Jaenisch, “Pluripotency and cellular reprogramming: facts, hypotheses and unresolved issues”, Cell 143:508-25 2010.

Sui Huang Rudolf Jaenisch

dynamics on a potential surface

tri-stability

dynamics in the state space – steady states

the first thing to calculate are the steady states because they are the skeleton around which the dynamics takes place

x1

x2

a

b

positive feedback by a gene on itself – a potential way to create bistability

the x1 nullcline is the locus of points satisfying

method of nullclines for 2D systems

the x2 nullcline is the locus of points satisfying

the steady states are the intersections of the two nullclines

x2

x1

x2

x1

STATE SPACES

how do we determine the stability of the steady states?

synthesis rates

degradation ratesweakness of

positive feedback

stability for 1D systems

1D dynamical system

f(x)

x

stableunstable

cannot tell how large the stability region is from the stability of the steady state

the awkward case

stableunstable

mixed

the 1D stability theorem

for any 1D dynamical system

1. find a steady state x = xst , so that

2. calculate the derivative of f wrt x at the steady state

3. if the derivative is negative then xst is stable

4. if the derivative is positive then xst is unstable

5. if the derivative is zero then nothing can be said

the nD stability theorem

for any n-D dynamical system

1. find a steady state x = xst , so that

2. calculate the Jacobian of f at the steady state

3. if all the eigenvalues of A have negative real part then xst is stable

4. if none of the eigenvalues has zero real part and one of them has a positive

real part then xst is unstable

5. if one of the eigenvalues of A has zero real part then nothing can be said –

the steady state is BAD (“non-hyperbolic”)

Jacobian matrix

the equivalent of the derivative wrt x is the JACOBIAN MATRIX of partial derivatives

linear approximation

nonlinear dynamics linear dynamics

mapping

position at time tmapping

position at time t

mapping

Hartman-Grobman Theorem – provided the steady state is GOOD (ie: “hyperbolic”), there is a continuous mapping from the nonlinear state space to the linear state space which preserves the dynamical trajectories,

x y

xst

a stability theorem for auto-regulation

assume general transcription & translation functions, linear degradation and arbitrary (positive or negative) feedback

x1

x2

nullcline geometry determines stability

x1 nullcline, in the 1st quadrant, crosses above x2 nullcline, in the 1st or 4th quadrants

STABLE

UNSTABLE

x1 nullcline, in the 1st quadrant, crosses below x2 nullcline, in the 1st quadrant

see the “nullcline theorem” handout for details

x1

x2

x1

x2

x1

x2

x2

x1

x2

x1

positive auto-regulation of a single gene

only a single stable steady state – if the “on” state, with positive x1 and x2, is stable (right), then the “off”state is unstable

how do we make bistability with the “off”state and the “on” state both stable?

implementing bistability through auto-regulation

positive feedback has to be combined with a sigmoidal (“S-shaped”) nullcline to create a threshold

a Hill function GRF is not essential for bistability (although often assumed – as we discussed in Lecture 8) but sharper switching can increase the bistable region in parameter space

x2

x1

Hill functiona more

realistic GRF

testing bistability by hysteresis

stead

y s

tate

x2

the switch between “low” and “high” (on/off) takes place at different values of the control parameter, depending on the starting state and the direction of change

change parameter slowly (“adiabatically”)

parameter

bistable systems exhibit history dependence to changes in parameters

low/high

high/low

decreasing

x2

x1

in practice

Pomerening, Sontag & James Ferrell “Building a cell cycle oscillator: hysteresis and bistability in the activation of CDC2”, Nature Cell Biol 5:346-51 2005

Sha, Moore, Chen, Lassaletta, Yi, Tyson & Sible, “Hysteresis drives cell-cycle transitions in Xenopus laevis egg extracts”, PNAS 100:975-80 2003

Ozbudak, Thatai, Lim, Shraiman & van Oudenaarden, “Multistability in the lactose utilization network of Escherichia coli”, Nature 427:737-40 2004

Isaacs, Hasty, Cantor & Collins, “Prediction and measurement of an autoregulatory genetic module”, PNAS 100:7714-9 2003

bifurcations – local, co-dimension one

1. a single real eigenvalue goes through 0

the real part of an eigenvalue of the Jacobian goes through 0

xx

x

x

x

x

eigenvalues of the Jacobian in the complex plane

there are three normal forms for this

local – near a steady state; co-dimension one – involving one parameter only

symmetric under conjugation

saddle-node

transcritical

k < 0 k = 0 k > 0

pitchfork (supercritical)

in the vicinity of the bifurcation, and in the vicinity of the steady state, the dynamics is given approximately by one of the following forms

mutual annihilation

stability exchange

spawning

normal forms

2. a pair of complex, conjugate eigenvalues cross the imaginary axis

the Hopf bifurcation

x x

x

x

x

x

eigenvalues of the Jacobian in the complex planeSelkov model (*)

(*) Strogatz, Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry and Engineering, Westview Press 2001

limit cycle

pluripotency in embryonic stem (ES) cells

Pan, Thomson, “Nanog and transcriptional networks in embryonic stem cell pluripotency”, Cell Res 17:42-9 2007; Chambers et al, “Nanog safeguards pluripotency and mediates germline development”, Nature 450:1230-4 2007

mouse ES cells, GFP expression from Nanog locus

weak linkage by variation/selection

excitability

there is a single, stable steady state with a small stability region, outside of which trajectories make long excursions before returning to the steady state

Kalmar et al, “Regulated fluctuations in Nanog expression mediate cell fate decisions in embryonic stem cells”, PLoS Biol 7:e1000149 2009; Eugene Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, MIT Press, 2007

monostability excitability oscillation

not potential dynamics!