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Supporting Information Text S2: Mathematical model for pseudo-somatic cells.
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Here we consider a multicell in which all cells are germ, but they differ in function performance. For
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simplicity, we shall assume that every cell falls into one of two states: an active “pseudo-soma” state or
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a relatively quiescent “true-germ” state. Ignoring development (growth and differentiation of cells), we
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consider the multicell starting with n cells. Suppose that a fraction f of these cells are pseudo-soma and
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(1- f ) are true germ cells. Each time step, we imagine that every cell performs functions (very few for
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the true germ and many for the pseudo-soma). The functions performed by pseudo-soma are mutagenic
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and destroy the performing cell with some probability. Thus, the number of functional pseudo-soma
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cells can decrease over time. Let Pt and Qt be variables to refer to the number of functional pseudo-
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soma cells and quiescent germ cells at time t , respectively. Assume that each functional pseudo-soma
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cell and quiescent germ cell contributes rP Ptg -1 and rQQtg -1 resources to the multicell at time step t .
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Here r P and rQ give the amount of resources contributed by a single isolated pseudo-soma cell and
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single isolated germ cell, respectively. We assume rP >rQ , that is, more resources are gathered by
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pseudo-soma cells than germ cells. The parameter g controls the concavity of the functional relationship
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between resource acquisition rate per cell and the number of cells. We assume 0 <g < 1, that is, resource
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acquisition per cell decreases with the number of cells performing functions. Such density dependent
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resource acquisition is necessary for evolving digital multicells that perform a diverse suite of functions
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(see Table S5 for further details). The accumulated resources in the multicell at time T is:
T -1
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RT = å ( rP Ptg +rQQtg ).
[S1]
t=0
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Because a quiescent cell is immortal, Qt = n(1- f ) for all t . The number of functional pseudo-soma
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cells (Pt ) is expected to decrease over time. For simplicity, we will assume that a fraction  will
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experience a destructive mutation. Thus, Pt  nf (1   )t .
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In order to replicate, suppose that the multicell requires R* resources. Then, by equation [S1], the T
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value that solves the following equation gives the time of multicell replication:
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ì T -1
ü
R* = rP ng f g íå (1- m )g t ý +rQ ng (1- f )g T .
î t=0
þ
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Letting r = rP / rQ and rQ* = R* / (ng rQ ) and simplifying equation [S2] yields:
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rQ* = r f g
1- (1- m )g T
+ (1- f )g T .
1- (1- m )g
[S2]
[S3]
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Let T * be the value of T that solves equation [S3]. Although there is no analytical solution for .., one
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can solve for its value numerically. For simplicity, we shall assume that after a multicell attempts
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reproduction, the parent is “reset” to its native state with all pseudo-soma cells functional.
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reproductive attempt is successful if a quiescent germ cell or a functional pseudo-soma cell is chosen as
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the propagule. Given that we know the time of multicell reproduction (T * ) , the rate of growth of the
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population of multicells can be calculated as:
The
1
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l = {2 - f (1- (1- m )T )} T .
*
*
[S4]
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We let fopt = argmax f (l ) . In Supporting Information Figure S3, we show fopt as a function of m and
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r . As pseudo-soma functions become less mutagenic ( m decreases) or as the resource acquisition rate
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of pseudo-soma cells relative to true germ cells increases ( r increases), the optimal fraction of pseudo-
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soma increases. We note that a large portion of parameter space has 0 < fopt < 1; thus, a type of division
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of labor is optimal in this simple model.
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