1. Denis Tverskoi1,2, Vladimir Makarenkov3, Fuad Aleskerov1,2
Modeling functional specialization of a cell colony under different fecundity and viability rates and a resource constraint
Denis Tverskoi1,2, Vladimir Makarenkov3, Fuad Aleskerov1,2
1International Laboratory of Decision Choice and Analysis, National Research University Higher School of Economics (HSE), Moscow, Russian Federation
2V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences (ICS RAS), Moscow, Russian Federation
3Département d’Informatique, Université du Québec à Montréal, Montréal, Canada
BIOMAT 2017, November 02

2. Volvocalean Green Algae
Michod RE, Viossat Y, Solari CA, Hurand M, Nedelcu AM. Life-history evolution and the origin of multicellularity. J Theor Biol. 2006;239: 257–272.

3. Full optimization model
vi – viability-enhancing capability of cell i. bi – resulting contribution of cell i to the fecundity of the group. Fitness trade-off function: Group’s level of fecundity and group’s level of viability: The group fitness:

4. Full optimization model
Main assumption: All cells in the colony suppose to be identical, so they have the same intrinsic trade-off functions.

5. Results
If trade-off functions are strictly concave, then the group of cells should remain unspecialized.
If trade-off functions are linear , then the group of cells behaves as if there was just one cell; therefore, each cell is indifferent to specialization.
If trade-off functions are strictly convex, then the group of cells aims for full specialization. In addition, if there is an even number of cells in the group, then half should specialize in germ and a half in soma. If there is an odd number of cells in the group, [N/2] of these cells should specialize in germ, [N/2] should specialize in soma and one cell should remain unspecialized.

6. Disadvantages of Full optimization model:
Assumption that all cells in the colony should be identical.
Full optimization model does not strictly reflect the fact that some environmental factors may influence the fitness of the colony.
The aim of this work:
We attempt to construct natural model of a colonial organism based on Full optimization model and we seek to understand incentives for specialization within this colony depending on a given fitness function encompassing different fecundity and viability rates and a specific resource constraint.

7. Optimization model for the colony of genetically identical cells
- Consider a colony of N cells
- Let bi be fecundity of the cell i and vi be viability of this cell
- Trade-off functions can be mathematically represented as follows:
the trade-off function should satisfy a set of requirements:
1. Let 𝑏 𝑚𝑎𝑥 ∈𝑅, 0< 𝑏 𝑚𝑎𝑥 <∞;
2. 𝜑:[0, 𝑏 𝑚𝑎𝑥 ]→𝑅, 𝜑∈ ℂ [0, 𝑏 𝑚𝑎𝑥 ] 2 ;
3.𝜑 0 = 𝑣 𝑚𝑎𝑥 , 0<𝑣 𝑚𝑎𝑥 <∞;
4. 𝜑 𝑏 𝑚𝑎𝑥 =0;
5. ∀𝑖= 1,𝑁 ∀ 𝑏 𝑖 ∈[0, 𝑏 𝑚𝑎𝑥 ] : 𝑑𝜑 𝑏 𝑖 𝑑 𝑏 𝑖 <0.

8. Optimization model for the colony of genetically identical cells
The group’s level of fecundity, B and the group’s level of viability, V are the following:
We suggest using the following fitness function:
Parameters α and β reflect the “importance” of the fecundity and viability contributions correspondingly to the fitness of the colony

9. Optimization model for the colony of genetically identical cells
A number of empirical works have shown that there exists a relationship that links fecundity of the colony, its viability and the environmental factors: We will further consider the parameter C as the amount of resources available to the colony. For the sake of simplicity, we assume that the function f has the following linear form:

10. Optimization model for the colony of genetically identical cells
- Our objective is to find the set of all optimal fitness strategies for a given colony and make conclusions about its specialization. - The cell i specializes in soma if and only if vi = vmax, and the cell i specializes in reproductive function (germ) if and only if bi = bmax.

11. General results
There are three possible cases when looking for the solution of the optimization problem under study. Case 1. The resource constraint does not influence the colony’s well-being. This means that we have a resource restriction such that an optimal fitness strategy for the colony without any resource restriction continues to be available to the colony regardless of environmental constraints.

12. General results
Case 2. Consider the following set of fecundity values: This case implies that this set is not empty. Case 3. The third case implies that the resource constraint influences the colony’s well-being and the set A is empty.

13. Convex trade-off function
In all panels, k1 = k2 = 1, α = 1 and φ = (b-1)2. In panel (a), β = 0.5 and C=2.85. In panel (b), β = 2 and C=2.85. In panel (c), β = 4 and C=2.85. In panel (d), β = 4 and C=2.5. Optimal strategies of the colony are colored in red.

14. Linear trade-off function
In all panels, α = 1, β = 2, k1 = 1, C = 4 and φ =1-b. In panel (a), k2 = 1. In panel (b), k2 = 2. Optimal strategies of the colony are colored in red. With a linear trade-off, a genetically identical colony behaves as a single cell.

15. Concave trade-off function
In all panels, α=1 and φ =1-b2. In panel (a), β = 0.5, C=5.5, k1 = 2 and k2 = 1. In panel (b), β = 2, C = 5, k1 = 2 and k2 = 1. In panel (c), β = 0.25, C=5, k1 = 2 and k2 = 1. In panel (d), β = 1, C = 3.125, k1 = 1 and k2 = 1. Optimal strategies of the colony are colored in red.

16. Optimization model for the colony of cells of different types
Here we assume that some cells of the colony are not equivalent with respect to a task (viability or fecundity). There are several reasons that can lead to the emergence of this non-equivalence:
Mutations
Positional effects

17. Results: Case 1, convex trade-offs
An optimal strategy b* implies that all cells, or all cells except one cell, should be specialized and that the number of cells specialized in germ (or soma) depends on cells positional effect and on the ratio (α/β).
In all panels, α = 1, φ1 = (b1-2)2, φ2 = (b2-2)2 and φ3 = (b3/√3 - √3)2. In panel (a), β = 2. In panel (b), β = 0.5. Optimal strategies of the colony are colored in red.

18. Results: Case 1, linear trade-offs
In optima there can be at most one unspecialized type of cells.
In all panels, α = 1, φ1 = 4-b1, φ2 = 4-b2 and φ3 = 2-b3/3. In panel (c), β = 2. In panel (d), β = 0.5. Optimal strategies of the colony are colored in red.

19. Results: Case 1, concave trade-offs
- There exists the unique optimal strategy. - Specialization emerges if and only if there does not exist a point b*, such that for all i = 1…N: 0 < bi* < bimax and
In all panels, α = 1 ), φ1 = 4-b12, φ2 = 2-b22/8 and φ3 = 3-b32/3. In panel (e), β = 2. In panel (f), β = 0.5. Optimal strategies of the colony are colored in red.

20. Conclusion
We have presented and analyzed a new general mathematical model for studying the emergence of germ-soma specialization. We have examined how the division of labor in a cell colony depends on the shape of trade-off functions, a resource constraint and different fecundity and viability rates.
We have shown that significant resource restrictions and a trade-off between the production difficulties and the fitness benefits of fecundity can cause the emergence of specialization in small-sized colonies.
We have shown that positional effects and mutations in small-sized colonies can lead to the emergence of soma specialization without changes in size or in resource restrictions.

21. Thank you for attention