Lateral Inhibition through Delta-Notch Signaling: A Piecewise Affine Hybrid Model

1. Lateral Inhibition through Delta-NotchSignaling: A Piecewise Affine Hybrid Model RonjoyGosh Claire Tomlin

2. AGENDA • Delta Notch Signaling • Hybrid Automata • What is Hybrid Automata • Authors Model • Algorithm • Query based technique for understanding reachable sets • Results/Conclusion • Future Work

3. INTRODUCTION • Understand the development of human embryo • How diseases develop • The authors’ research focuses on a specific biological mechanism known as intercellular protein signaling. • Intercellular signaling is a feedback network which interrelates the fate of a single cell and its neighbors • Delta-Notch signaling mechanism, responsible for pattern formation in many different biological systems, such as the emergence of ciliated cells in Xenopusembryonic skin

4. INTRODUCTION • A small number of signaling pathways are used iteratively to regulate cell fates, cell proliferation and cell death in development. • Notch is the receptor in one such pathway • Ligands are also trans membrane proteins therefore signaling is restricted to neighboring cells • The intracellular transduction of the Notch signal is remarkably simple, with no secondary messengers, this pathway functions in an enormous diversity of developmental processes and its dysfunction is implicated in many cancers

5. What is Hybrid Automata • When we need to be able to describe systems that contain both the discrete switching logic and continuous dynamics • Hybrid Automata = Finite state machines(discrete logic) that continuously changes (continuous dynamics)

6. Hybrid Automata • Let x be the state of the system • As we will be switching between different modes of operation. Lets add an additional discrete state q • Now the dynamic can be encoded as: ẋ = fq (x,u) ẋ = fq ẋ = fq’ q q’

7. Hybrid Automata • The boundary condition or threshold under which a transition occurs is called the “guard condition” • That is the transition from discrete state q to q’ x εGq,q’ • Another key component is called Resets(R) • This factors accounts for the abrupt change that occurs during the transition in the state. x := Rq,q’ (x)

8. The Hybrid Automata Model Example

9. Why Hybrid Automata • Biological cell networks exhibit complex combinations of both discrete and continuous behaviors • The dynamics that govern the spatial and temporal increase or decrease of protein concentration inside a single cell are continuously changing • While the activation or deactivation of these continuous dynamics are triggered by discrete switches when protein concentrations reach certain thresholds. • The authors model a hybrid system which has a striking resemblance to the behavior in a biological mechanism called Delta-Notch signaling. • Intercellular signaling is a feedback network which interrelates the fate of a single cell and its neighbors

10. Why Hybrid Automata • Deltais a transmembrane protein that binds and activates its receptor, the transmembrane protein Notch, in neighboring cells. • The activation of Notch has a direct and immediate effect on gene expression. • Hence Notch signaling directly controls switching in genetic networks and cascades.

11. Why Hybrid Automata • The protein concentration dynamics inside each biological cell are modelled using linear differential equations • Inputs activate or deactivate these continuous dynamics through discrete switches • Which themselves are controlled by protein concentrations reaching given thresholds • Hybrid automata theory presents an ideal framework model and analyze these processes

12. Why Hybrid Automata • Using simple continuous dynamics and lumping the complexity into the discrete inputs gives us the capability to analyze the model mathematically • Helps to prove reachability and convergence for a wide set of initial conditions • Vary and extract important parameters and predict their effects on the system evolution without simulation • This can be used to suggest biological experiments to validate the model as well as refine it.

13. Design of the Model • The following properties, based on experimental data, are incorporated in the model: • Direct contact between cells is a prerequisite for Delta-Notch signaling to occur • Notch production is triggered by high Delta levels in neighboring cells • Delta production is triggered by low Notch concentrations in the same cell • High Delta concentrations lead to differentiated cells and low Delta levels to undifferentiated cells • Both Delta and Notch proteins decay exponentially

14. Design of the Model • In the model, the cells are assumed to be hexagonal close packed, i.e. each cell has six neighbors in contact with it

15. Where x1 and x2 is the Delta and Notch Concentrations • xDelta,i Delta protein concentration in ith neighboring cell • λD and λN Delta and Notch protein decay constants • RD and RN Delta and Notch protein production rates • hD and hNSwitching thresholds for Delta and Notch production • In single cell xDelta,i = 0,ε{1,….6} as there are no neighbors whose Delta protein can be sensed

16. Transition Diagram

17. Equilibrium Analysis • No two cells with high Delta protein concentrations can lie next to each other • Cell with high Notch protein concentration must have at least one neighbor with high Delta protein concentration • Cell with high Delta protein concentration has low Notch protein concentration and vice versa • By observation it is deduced for biological feasible equilibria to exist the following constraint is constructed: hD , hN : = RN/λN < hN≤ 0 ᴧ 0 < hN ≤ RD/λD

18. Equilibrium Analysis

19. Reachability: Query Based Interpretation • The key idea here is to query the computed reachable sets with logical expressions that encode biologically interesting conditions involving protein concentrations • This model will help in • Find steady states that can be attained from those conditions • Deduce any additions constraints on protein concentrations that are necessary to reach those steady states

20. Algorithm Input: A set of states, Reach = Inv(q1) V Inv(q2) V Inv(q3) ….. V Inv(qj) backward reachable only from state qk and a query statement Query Output: A logical statement Output for each disjunctInv(qi) Reach, I = 1,2, … j compute fi = Query ᴧ Inv(qi) return output = Vji=1fi

21. Algorithm Cont… • Query is a logical statement that involves protein concentrations. Example Query = Conca –Concb >Conccᴧ Concd< Concb where Conci is concentration of protein species i • Query ᴧ Reach: the expression used to find the truth value. This is an efficient computation because Reach is disjunctive form. • Hence we can use law of distributivuty • Query ᴧ Reach ≡ (Query ᴧ Inv(q1)) V (Query ᴧ Inv(q2)) V (Query ᴧ Inv(q3)) V …… ..(Query ᴧ Inv(qk)) • This implies that each of the disjuncts can be simplified separately

22. If the resultant expression is false then the query is not valid for that reachable set. • If the resultant expression is true then it always leads to particular steady state whose background reachable set was queried • Since each disjunct is evaluated separately the computation algorithm is highly parallelizable • All the computations is performed on the 4 biological feasible equilibria given in the table below

23. Example • In the above table it has 8 continuous state variables x1,x2, x3, …. ,x8 • x1,x3, x5, and x7 denote the Delta protein levels • x2, x4, x6, and x8 denote the Notch protein levels • The unique backward reachable sets for equilibrium 1,2,3 and 4 contain 780, 1565, 1566 and 920 states • Each state is defined by 8 inequality invariants • Hence the size of reachable sets are quite large and difficult to interpret by inspection

24. Example Cont... • Most of the simple queries tried by the authors took more than 2 hours to verify in 2GHz Pentium 4 Workstation • Query1: x1 > x3ᴧx1> x5ᴧx1 > x7 • Delta protein centration in cell 1 is higher than the Delta concentrations in the other 3 cells • This query tests the classical inhibition property i.e. amplification of initial protein concentration differences • Higher Delta protein concentrations in cell 1 is expected to suppress Delta protein growth in all its neighbor • Query 1 Results: • Equilibrium 1: Unreachable • Equilibrium 2: Unreachable • Equilibrium 3: Unreachable • Equilibrium 4: ( x3 – x1 < 0 ᴧ x5 – x1 < 0 ᴧ x7 – x1 < 0 ᴧ hD + x6 ≥ 0 ᴧ hD + x8≥ 0 ᴧhD + x4≥ 0 ᴧhD + x4≥ 0 ᴧ hD + x2 ≤ 0 ᴧ hN – 2x7 – 2x5 – 2 x3 ≥ 0 ᴧ hN – 2x7 – 2x5 – 2 x1≤ 0 ᴧ hN – 2x7 – 2x1– 2 x3 ≤ 0 VhN – 2x7 – 2x5 – 2 x1≥ 0 v (hD + x4 ≤ 0 ᴧ hN– 2x7 – 2x5 – 2 x3 >0) v

25. Example Cont... • The query result confirms that the fundamental inhibitory property of the network modeled faithfully by the hybrid automaton model • Only the backward reachable set of equilibrium 4 satisfies query 1 • The reachable sets for the other 3 equilibria do not • For example the total Delta protein concentration in cells 2,3 and 4 is required to be less than half the switching threshold value for turning on Notch production

26. Conclusions • Gives a glimpse of the immense opportunities for hybrid modeling in biological systems • The preliminary analysis of the model is promising and has a resulted in the identification of the threshold parameters, which is an important and arbiter of cell fate • Predications can be made using this model to regarding the various initial states of the cell • The clear limitation of this approach is the complexity in the dimension of the continuous state • Future work concentrates on the development of automated tools for analyzing equilibrium and convergence

27. Thank You