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!
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