George Kalosakas University of Patras, Materials Science Dept.

1. Biological patterns: Lateral inhibition through the Notch-Delta system George Kalosakas University of Patras, Materials Science Dept. CCQCN, Physics Dept., Univ. of Crete

2. “Checkerboard” patterns in living organisms In many tissues fine-scale spatial patterns appear, where individual cells exhibit distinct behavior than their neighboring cells At an early stage of development all these cells are similar, but at a certain point some cells differentiate towards a particular fate (primary fate), expressing a specific genetic program Through proper signaling, its neighboring cells are inhibited to adopt the same fate, expressing different genetic program leading to a secondary fate Phillips, Kondev, Theriot, Garcia, Siepmann, Physical Biology of the Cell, 2nd Ed., Garland Science (2013)

3. Lateral inhibition Lateral inhibition provides a contact mechanism resulting in “checkerboard” patterns In lateral inhibition one cell adopting a primary fate communicates its decision to the adjacent cells, preventing them for doing the same The Notch-Delta system is a well studied example of lateral inhibition Both Notch and Delta are transmembrane proteins Notch increased activity inhibits the primary cell fate Notch acts as receptor activated by Delta (ligand) of an adjacent cell Within a cell, Notch activation suppresses Delta activity. Inactive Notch enhances Delta activity.

4. A feedback mechanism amplifying small differences Notch-Delta intercellular communication results in a feedback mechanism self-amplifying small differences of Notch activity (due to random fluctuations) between adjacent cells → different cell fates Decreased Notch activity (lower cell) → stimulation of Delta activity in the same (lower) cell Increased Delta activity (lower cell) → stimulation of Notch activity in the adjacent (upper) cell Increased Notch activity (upper cell) → deactivation of Delta activity in the same (upper) cell Decreased Delta activity (upper cell) → further inactivation of Notch in the adjacent (lower) cell

5. A simple model for the Notch-Delta system Ni and Di are the levels of Notch and Delta activity in the ith cell (normalized to their maximum values) γN and γD are rate constants describing the decay of Notch and Delta activities Dia denotes the average level of activity of the adjacent cells F(x) is an increasing function representing the activation of Notch in the ith cell by the Delta activity of adjacent cells G(x) is a decreasing function representing the inactivation of Delta in the ith cell by the Notch activity of the same cell Collier, Monk, Maini, Lewis, J. Theor. Biol. 183, 429 (1996)

6. Experimental determination of Notch activity as a function of Delta signal Time-lapse video of YFP activation Initially DAPT inhibits Notch activation. DAPT is removed at t=0 Cells placed on a surface with a uniform concentration Dplate of Delta. Upon binding to Delta, Notch activates expression of yellow fluorescent protein gene (YFP) YFP fluorescence for different Delta concentrations on the plate Notch activity (YFP production rate) as a function of Delta concentration Sprinzak, Lakhanpal, LeBon, et al., Nature 465, 86 (2010)

7. A dimensionless form of the system Dividing both equations by γN we get τ = γN t is a dimensionless time f(x) =F(x) / γN g(x) =G(x) / γD v = γD / γN is the ratio of the decay rates (measure of relative time scales for Notch and Delta activity variation)

8. Two cell system Steady states: (N1*, D1*, N2*, D2*) = (N1*, g(N1*), N2*, g(N2*)) with N1*=fg(N2*) and N2*=fg(N1*) fg is monotonic decreasing function → there exists only one x0ϵ [0,fg(0)] such that fg(x0)= x0 There is exactly one homogeneoussteady state (N1*, D1*, N2*, D2*) = (x0, g(x0), x0, g(x0)) Linear stability analysis shows that this homogeneous steady state is stable when f ′(g(x0)) ∙ g ′(x0) > -1 and unstable when f ′(g(x0)) ∙ g ′(x0) < -1 There may be one or more pairs of heterogeneous steady states A period 2 solution (x1,x2) of the map zi+1= fg(zi) satisfies x1= fg(x2), x2= fg(x1), and x1≠x2 This gives a pair of heterogeneous steady states (N1*, D1*, N2*, D2*) = (x1, g(x1), x2, g(x2)) or (x2, g(x2), x1, g(x1)) There is at least one such solution whenf ′(g(x0)) ∙ g ′(x0) < -1

9. Two cell system: numerical simulations Parameters with unstable homogeneous steady state and a pair of heterogeneous steady states c′=0.01, c′′=100, v=1, zero boundary conditions Initial conditions: N1(0)=1 D1(0)=1 N2(0)=0.99 D2(0)=0.99 Cell 1 adopts primary fate Cell 2 adopts secondary fate

10. Two cell system: phase portraits for v >> 1 When v = γD / γN = τN / τD >>1, the values of Delta activity adjust faster to the instantaneous Notch activities, while Notch dynamics evolves more slowly Then we can consider D1 ≈ g(N1) and D2 ≈ g(N2), eliminating Delta variables We end up to a system with 2 degrees of freedom, that can be visualized using the phase portrait (depicting the direction of the vector (dN1/dτ, dN2/dτ) at each point of phase space) Red line: N1 = f(g(N2)) Blue line: N2 = f(g(N1)) The cell with the lower (higher) Notch activity initially adopts the primary (secondary) fate

11. Two cell system: phase portraits for v << 1 When v = γD / γN = τN / τD <<1, the values of Notch activity adjust faster to the instantaneous Delta activities, while Delta dynamics evolves more slowly. Then we can consider N1 ≈ f(D2) and N2 ≈ f(D1), eliminating Notch variables We end up to a system with 2 degrees of freedom, that can be visualized through some trajectories in phase space. The cell with the higher (lower) Delta activity initially adopts the primary (secondary) fate Only one steady state exists There exist three steady states

12. One-dimensional array of cells: numerical simulations Parameters with unstable homogeneous steady state and a pair of heterogeneous steady states c′=0.01, c′′=100, v=1, zero boundary conditions Initial conditions: Ni(0) as in top graph (t=0), Di(0)=1 The final pattern appears to be one of alternating low and high levels of Notch activity (adopting primary and secondary fate, respectively) Defects can occur with two adjacent cells exhibiting high Notch activity

13. One-dimensional array of cells: linear stability analysis of the homogeneous steady state We consider small perturbations around the homogeneous steady state: ni = Ni - x0, di = Di - g(x0) Linearizing the equations of the system Looking for patterns with period N (N>1) in an infinite array, the linearized equations can be decoupled using the transform Then the linearized equations in the new variables give pairs of uncoupled linear equations with constant coefficients for each r=1,…,N

14. Solving the uncoupled linear system, we find the general solution where and are the eigenvalues and the eigenvectors of the matrix The homogeneous steady state is linearly stable if and only if which is equivalent to The eigenvalues with the largest real part are where N is even The dominant pattern for large t is When the instability criterion is satisfied, the homogeneous steady state is unstable to a solution with period 2 in space. Small perturbations grow exponentially on a time scale

15. Two-dimensional hexagonal array of cells: steady state patterns in numerical simulations The default s.s. pattern for fairly homogeneous initial conditions, Nij(0)≈1, Dij(0)≈1, when the parameters are such that the homogeneous steady state is unstable, is the “checkerboard” pattern Part of a 6x6 array with periodic boundary conditions c′=0.01, c′′=100, v=1 Other s.s. patterns with defects can also arise, when the boundary conditions or the size of the system is not compatible with such pattern 8x8 array with zero boundary conditions 7x7 array with periodic boundary conditions

16. Two-dimensional array of cells: linear stability analysis of the homogeneous steady state Small perturbations around the homogeneous steady state: ni,j = Ni,j - x0, di,j = Di,j - g(x0) The linearized equations can be decoupled using the transform Then the linearized equations in the new variables give pairs of uncoupled linear equations with constant coefficients for each r=1,…,N and q=1,…,M The homogeneous steady state is linearly stable if and only if When the instability criterion is satisfied, the homogeneous steady state is unstable and small perturbations grow exponentially on a time scale

17. References Collier, Monk, Maini, Lewis, J. Theor. Biol. 183, 429 (1996) Phillips, Kondev, Theriot, Garcia, Siepmann, Physical Biology of the Cell, 2nd Ed., Garland Science (2013)