Detection of multi-stability in biological feedback systems
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Detection of multi-stability in biological feedback systems George J. Pappas University of Pennsylvania Philadelphia, USA
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Detection of multi-stability in biological feedback systems. George J. Pappas University of Pennsylvania Philadelphia, USA. The paper. Bio Bi-stability. Even simple signaling networks result in bi-stability Bistability: Toggling between two alternative steady state - PowerPoint PPT Presentation
Transcript of Detection of multi-stability in biological feedback systems
Detection of multi-stabilityin biological feedback systems
George J. Pappas University of Pennsylvania
Philadelphia, USA
The paper
Bio Bi-stabilityEven simple signaling networks result in bi-stability
Bistability: Toggling between two alternative steady state
Reasons/uses for bi-stability
Switch-like biochemical responsesMutual exclusive cell cycle phasesProduction of biochemical memoriesRapid propagation of receptor activation
Bi-stability(a) arises in systems with positive feedback loops
(b) mutually inhibitory, double negative feedback,(c) Realistic biological networks with positive/negative
feedback
State-of-the-art in detecting multi-stability
Positive feedback is necessary but not sufficient
Graphical phase analysis available for 2D systems
Game plan
2D example : Cdc2-Cyclin B/Wee1 System
Develop framework for detecting multi-stabilityShow modularity and scalabilty of approach
5D example : Mos/MEK/p42 MAPK Cascade
Cdc2-Cyclin B/Wee1 (two-protein) system
Assumptions: Each protein exists in only two forms Active/inactive Cdc2 (variables x1,x2) Active/inactive Wee1 (variables y1,y2)
Inhibition is approximated by a Hill equation
Cdc2-Cyclin B/Wee1 (two-protein) system
Assumptions: Each protein exists in only two forms Active/inactive Cdc2 (variables x1,x2) Active/inactive Wee1 (variables y1,y2)
Inhibition is approximated by a Hill equation
Cdc2-Cyclin B/Wee1 (two-protein) system
Constants : Rate constants alpha, betaKs are the Michaelis (saturation) constantsgamma are the Hill coefficientsv is the gain (strength) of Wee1 on Cdc2
Cdc2-Cyclin B/Wee1 (two-protein) system
X1+X2=1Y1+Y2=1
Cdc2-Cyclin B/Wee1 (two-protein) system2D phase plane analysis
Approach does not scale to higher dimensions
Key idea : Break the feedbackFeedback system
breaking the feedback results in open loop system
Main idea : Infer properties of closed-loop by open-loop
Re-closing loop=n= v y1
Rough Theorem (from open to closed loop)
Assume the open loop system
satisfies two critical properties
A) (Well-defined Steady State) For every constant input, there is a unique steady state response.
B) (I/O Monotone) There are no possible negative feedback loops, even when the system is closed under positive feedback
then the closed loop system
A) has three steady statesB) Almost all trajectories converge to one of two attracting
equilibria
Property A : Well-defined Steady StateFor the open loop system
we must have that for every constant (unit step) input, there is a unique steady state
Red curve in figure below(only output y1 is shown)
=
Note that this may be a hardthing to do!
Property B : I/O MonotonicityMain idea : Use the (directed) incidence graph of the system
Important : Effect of one variable on another must have the same sign globally. Otherwise their result does not apply.
For example, w affects derivative x1 in a globally decreasing manner.
Self-loops (for example –ax1 decay) are not included in the graph
Property B : I/O MonotonicityMain idea : Use the (directed) incidence graph of the system
Path Sign : Sign of a path is the product of the signs along the way
Monotonicity property(i) Every loop in the graph, directed or not, is positive(ii) All paths from input to output are positive(iii) There is a directed path from input to all states(iv) There is a directed path from all states to output
Application of main resultMain idea : Use the (directed) incidence graph of the system
Both properties have been verified. The potential equilibria are at
intersection of sigmoidal red curve and line Stable : Red curve slope < 1Unstable : Slope > 1
Hill coefficient > 1 importantBistability needs cooperativity
Almost all trajectories convergeto one of the stable equilibria
Hysteresis explained, using I/O methods
Monotonicity is necessaryConsider the Predator-prey like open loop system
satisfies one critical property (no monotonicity)
A) (Well-defined Steady State) For every constant input, there is a unique steady state response.
then the closed loop system
A) has multiple steady statesB) Almost all trajectories converge to one of two attracting
equilibria
Claim is false – System not monotoneConsider the Predator-prey like open loop system
Then a similar analysis results in
No global bi-stabilityLimit cycles exist
Modularity, scalabilityKey result : Cascade (series) composition of monotone systems
is monotone !
Therefore, multi-stability analysis of large biological networks, can be
deduced from analysis of smaller networks.
The Mos/MEK/p42 MAPK Cascade
A 5D case studyThe Mos/MEK/p42 MAPK Cascade
After 7D modeling and elimination of 2 conserved quantities, we get
After feedback breakup, this isa cascade composition of one1D and two 2D systems
Property A and B are composable.Thus 5D systems satisfies conditions.Hence system is multi-stable
The computational story : How many ? Numerical simulations to determine the global critical function
For SISO systems, Figure c above is always planar
SummaryOutstanding paper : Outstanding systems paper with potential impact in biology/networks.
Pushes systems thinking and potentially bio results
More complicated inter-connections possible.
Many future directions to consider, from a systems pointMonotone, on the averageDensity functions for monotone systemsCompositions/decompositions to monotone systems
