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 Reasons/uses for bi-stability Switch-like biochemical responses Mutual exclusive cell cycle phases Production of biochemical memories Rapid 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-stability Show 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 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 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, beta Ks are the Michaelis (saturation) constants gamma are the Hill coefficients v is the gain (strength) of Wee1 on Cdc2 Constants : Rate constants alpha, beta Ks are the Michaelis (saturation) constants gamma are the Hill coefficients v is the gain (strength) of Wee1 on Cdc2

Cdc2-Cyclin B/Wee1 (two-protein) system X1+X2=1 Y1+Y2=1

Cdc2-Cyclin B/Wee1 (two-protein) system 2D phase plane analysis Approach does not scale to higher dimensions

Key idea : Break the feedback Feedback 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 states B) Almost all trajectories converge to one of two attracting equilibria

Property A : Well-defined Steady State For 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 hard thing to do! Note that this may be a hard thing to do!

Property B : I/O Monotonicity Main 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 Monotonicity Main 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 result Main 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 < 1 Unstable : Slope > 1 Stable : Red curve slope < 1 Unstable : Slope > 1 Hill coefficient > 1 important Bistability needs cooperativity Hill coefficient > 1 important Bistability needs cooperativity Almost all trajectories converge to one of the stable equilibria Almost all trajectories converge to one of the stable equilibria

Hysteresis explained, using I/O methods

Monotonicity is necessary Consider 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 states B) Almost all trajectories converge to one of two attracting equilibria

Claim is false – System not monotone Consider the Predator-prey like open loop system Then a similar analysis results in No global bi-stability Limit cycles exist No global bi-stability Limit cycles exist

Modularity, scalability Key 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 study The Mos/MEK/p42 MAPK Cascade After 7D modeling and elimination of 2 conserved quantities, we get After feedback breakup, this is a cascade composition of one 1D and two 2D systems After feedback breakup, this is a cascade composition of one 1D and two 2D systems Property A and B are composable. Thus 5D systems satisfies conditions. Hence system is multi-stable 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

Summary Outstanding 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 point Monotone, on the average Density functions for monotone systems Compositions/decompositions to monotone systems