understanding organism growth and cellular differentiation ......cellular automata (see [44][17]...
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Understanding Organism Growth and Cellular Differentiation Through
Evolutionary Selection of Dynamical Properties.
by
Yuri Cantor
Thesis Proposal for the degree of
Doctor of Philosophy
at the
Graduate Center of the City University of New York
Committee: Professor Bilal Khan (Advisor)Professor Nancy GriffethProfessor Matthew JohnsonProfessor Kirk Dombrowski
May, 2016
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Contents
1 Abstract 1
2 Introduction 2
2.1 Preliminary Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Area 1: Cellular Differentiation and Robustness 8
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 Formal Hypothesis 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.4 Research Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.5 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Area 2: Cellular Differentiation and Adaptivity 12
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.3 Formal Hypothesis 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.4 Research Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.5 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 Area 3: Symmetry and Growth 16
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.3 Formal Hypothesis 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.4 Research Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
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5.5 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6 Obstacles 24
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.2 Candidate Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.2.1 Powers of 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.2.2 Sampling techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.2.3 Distributed processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
7 Summary 27
8 Timeline 28
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1 Abstract
Computer simulations of networks of cellular automata serve as an experimental approach to explor-
ing relationships between properties of a network and the dynamics of the network. Consequently,
these dynamical systems are commonly used to model biological organisms representing cells with
the abstraction of cellular automata.
Natural selection describes a process where traits or properties of the biological organism that are
better suited to the environment will be selected for. It follows that properties of the dynamics that
are better suited to the environment will also be selected for. This proposal studies the impact of
Darwinian selection on cellular differentiation and organism growth and focuses on the relationship
of organism properties of homogeneity and heterogeneity to the dynamical properties of robustness,
adaptivity, and chromatic symmetry.
I propose an experimental approach to show that cellular differentiation increases the expected
robustness in an organism’s dynamics, cellular differentiation leads to increasing adaptivity, and
organism growth by elongation preserves symmetry in the dynamics of an organism. Further, I will
also address practical concerns arising from computational limitations and big data in experimental
approaches to studying dynamical systems.
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2 Introduction
Biological neural network behavior is difficult to model computationally. These networks are com-
posed of densely connected neurons, have a state of excitation that is continuous in degree, and
transmit state asynchronously [50]. There has been extensive research into biological networks and
cellular automata (see [44][17] for brief surveys). Cellular automata as described by Von Neumann
[40] can be connected to form a network which serves as an abstraction of a biological networks and
computational simulation of these simplified networks of cellular automata can lead to understand-
ing the behavior in correlated complex biological networks [28][33][46][19].
In an experimental approach abstractions are necessary to maximize the number of networks and
depth of simulation that can be modeled. Further, it is known that decomposing super networks into
sub networks can lead to understanding pathways and signaling in biological circuitry [53]. Random
Boolean Networks (RBNs) have been used to abstract from continuous state values to Boolean.
Kauffman’s NK network model abstracts from densely connected to K connections [20][21]. The
trivial case where K=1, restricts the nodes to connected along a line. While Conway’s "Game of
Life" is example of a Boolean NK network represented using a two dimensional grid where K=8
[18]. Asynchronous transmissions with small temporal tolerances can be modeled as synchronous
[39].
This proposal explores theoretical synthetic biology, using the more restricted class of synchronous
Boolean cyclic NK networks (where K=2) as a formal basis. This class of networks has received
considerable attention [47], and exhibits many of the phenomena seen in more general NK coun-
terparts [45]. Even with these reductions in complexity, fully simulating the dynamics remains
computationally intractable except for networks of relatively small size [51].
The relationship of network structural properties to dynamical properties is crucial to understand
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how and why biological networks have evolved. Further, this relationship can be used to construct
networks with desirable properties [41]. Dynamical systems have been previously be evaluated for
their landscape ruggedness [35][34], redundancy [21], and reversibility and surjectivity (reachability
and Garden of Eden states [47]) [26][37]. I propose an experimental approach to evaluate the
relationship between network structural properties of heterogeneity, homogeneity, and growth to
dynamical properties of robustness [9], adaptivity [7], and symmetry [10].
This proposal is divided into four parts. The first three parts correspond to three hypotheses
intended to provide insight to the relationship between network structural properties and dynamical
properties: cellular differentiation increases the expected robustness in an organism’s dynamics,
cellular differentiation leads to increasing adaptivity, and organism growth by elongation preserves
symmetry in the dynamics of an organism. The fourth part will address practical concerns to
the experimental approach arising from computational limitations and big data in experimental
approaches to studying dynamical systems.
2.1 Preliminary Definitions
Throughout this proposal networks of cellular automata will be referred to as organisms where
each node in a network will be a cell of the organism. The application of biological terminology to
describe theoretical and synthetic networks is common practice as seen since Von Neumman’s seminal
work on cellular automata [40][30]. As noted in the introduction the structure I am focusing on is
selected to decrease computational complexity in order to facilitate deeper exploration of state space
and exploration of larger organisms.
Structure. Informally, linear cyclic organisms are composed of cells that are directly connected
to their two neighboring cells such that the cells form a ring where an initial cell is connected to the
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Table 1: Truth table mapping with inputs at time t and resulting output at time t+ 1
s(vi�1, t) s(vi, t) s(vi+1, t) s(vi, t+ 1)
0 ⇤ 0 b00 ⇤ 1 b11 ⇤ 0 b21 ⇤ 1 b3
Table 2: XOR b0b1b2b3 = 0110 truth table mapping
s(vi�1, t) s(vi, t) s(vi+1, t) s(vi, t+ 1)
0 ⇤ 0 00 ⇤ 1 11 ⇤ 0 11 ⇤ 1 0
next cell and the last cell. Formally, the linear cyclic structure is modeled as an undirected cyclic
graph C = (V,E) of size n. Vertices are the cells, and are enumerated V = {v0, . . . , vn�1} where each
cell vi in V is connected in cyclic order to two neighbors such that edges E = {(vi, vi+1 (mod n))|i =
0, . . . , n � 1}. The set of possible linear cyclic organisms of size N are a subset of Kaufman’s NK-
networks [12] of size N where for each cell the number of inputs is K = 2.
Cell state is Boolean, either 0 or 1, and deterministic by fixing a function f : V �! F that
assigns to each cell v 2 V , a function f(v) from F = {g : {0, 1}{0, 1} �! {0, 1}}, the set of all
binary Boolean functions. The action of f at a vertex vi can be thought of as a truth table mapping
from the two inputs K, vi’s left and right neighbors’ current state, to vi’s state at the next time
step. This function differs from the traditional Wolfram model [51][55] in that Wolfram’s model
used K = 3 and considered the self-input or state of cell vi as well. The bits b0, b1, b2, b3, as in
Table 3, must be either 0 or 1 and the 4-bit binary string b0b1b2b3 is used to name the function f.
Note that because there are 2 choices for each of the 4 possible input a cell’s state must be 0 or 1
as a result of the possible the inputs from its neighbors, |F | = 222 = 16. For example, Table 2 is
the XOR function truth table. Note, that the space of possible functions for Wolfram’s model is the
more expansive set such that |F | = 223 = 256 [52]. In Table 3 these rules are defined using Boolean
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Table 3: Table of update rules.Rule numbers expressed as Boolean logic with 2 inputs:
The value of the left neighbor vi�1, and the value of the right neighbor: vi+1Rule Boolean Logic
Rule 0= 0Rule 1= ¬vi�1 ^ ¬vi+1Rule 2= ¬vi�1 ^ vi+1Rule 3= ¬vi�1Rule 4= vi�1 ^ ¬vi+1Rule 5= ¬vi+1Rule 6= (vi�1 ^ ¬vi+1) _ (¬vi�1 ^ vi+1)Rule 7= ¬(vi�1 ^ vi+1)Rule 8= vi�1 ^ vi+1Rule 9= (vi�1 ^ vi+1) _ (¬vi�1 ^ ¬vi+1)Rule 10= vi+1Rule 11= vi+1 _ (¬vi�1 ^ ¬vi+1)Rule 12= vi�1Rule 13= vi�1 _ (¬vi�1 ^ ¬vi+1)Rule 14= vi�1 _ vi+1Rule 15= 1
logic and ordered in ascending order according to the Boolean input values. For reference, rule 6 in
Table 3 corresponds to the earlier example truth table shown in Table 2.
The organisms in this proposal are synchronous where every cellular state is instantaneously
determined at each time interval according to:
s(vi, t+ 1) = f(vi)�s(vi�1 (mod n), t), s(vi+1 (mod n), t)
�
for each i = 0, . . . n�1 and t > 0. Note that results from synchronous models can be transformed and
be realized in asynchronous models with synchronization stays within small tolerances locally [39].
This implies that research into synchronous organisms has implications for asynchronous organisms.
Homogeneity and heterogeneity have previously been studied with respect to connectivity [32] and
functions [23]; however, in this proposal they will refer only to functions. If every cell in an organism
has the same function assigned to it, then the organism is homogeneous. There are 16 unique
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homogeneous organisms of any given size. However, if not every cell in an organism has the same
function assigned to it, then the organism is heterogeneous. There are 16n � 16 heterogeneous
organisms of size n. Formally, an organism is homogeneous if |Im(f)| = 1; otherwise it is hetero-
geneous. There has been research into the distance between functions using hamming distance as
a measure of proximity [27][55]. However, here I simplify and look only at the number of different
functions instead of the specificity of what those functions are and their distance from each other.
An organism state, as in Figure 3, is the ordered set of its cell states at time t. This set of
states for example in Figure 1, can be expressed as the sequence of states v0v1v2v3v4 = 00010.
Dynamics. The dynamics of the organism are represented as a directed graph S connecting
organism states, as in Figure 1, to their successor states, seen in Figure 2, representing the
organism’s phase space, in which (X, Y) is an edge if it can be said that whenever the organism
is in state X at time t, it is necessarily (absent noise) in state Y at time t + 1. Formally, the
dynamics is a directed graph S = (2V , D) whose vertex set consists of all possible states of the
organism (i.e. the power set of V), and whose edge set D includes every ordered pair (X, Y) for
which s+(t) = X =) s+(t + 1) = Y . If an organism state X is connected to organism state Y by
a directed edge from X to Y, then Y is the successor state of X. Not only have successor functions
have been studied for extensively, but their resulting dynamics and dynamical components have been
the focus of research problems [52].
0
1
00
0
Figure 1: Initial state.
1
0
10
0
Figure 2: Successor state.
v
v
vv
v
2
3
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1
Figure 3: Size 5 network.
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In the dynamics, a Garden of Eden state is a state that cannot be reached from any other state
by a directed edge in the dynamics space of an organism. In other words, no state in the dynamics
of the organism has as its successor a Garden of Eden state. This dynamical component has drawn
attention as part of the problem of reachability: to determine if a state is a Garden of Eden state
[47]. Research into Garden of Eden states involves developing algorithms for inverting the successor
function or reversibility [26][37], and Wuensche and Lesser introduced a reverse algorithm [59].
An attractor state is an organism state that occurs in a cycle in the dynamics space of an
organism. In other words, if X is an attractor state then there exists some discrete number of time
intervals j such that when starting at state X and time t and computing the successor states at
times t+1, . . . , t+ j then at the state at time t = t+ j will be X. An attractor is the set of all states
that are in the same cycle in the dynamics space of an organism and attractor length is the number
of states in an attractor. Here the term attractor encompasses the classes of attractors defined by
Wolfram including: fixed point, periodic, and strange [51]. In Hopfield networks attractors are the
content-addressable memory [25].
A tributary state is a state that is not in an attractor. When starting at a tributary state and
computing the successor states eventually an attractor state will be encountered because the state
space has a discrete number of possible states. A tributary is the set of states composing a path of
tributary states that lead to the first encountered attractor state when computing forward from a
Garden of Eden state. An attractor state can have many tributaries or no tributaries. Basins of
attraction [56] are formed by these tributaries.
An attractor with its tributaries composes a single connected component of the dynamics
graph of an organism. The graph of an organism’s dynamics space is made up of these components.
Figures 4, 5, 6 show an example decomposition of the size 12 homogeneous XOR organism. The
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complete dynamics graph for the homogeneous XOR organism of size 12 has 6 attractors of length
2 (Figure 4), 60 attractors of length 4 (Figure 5), and 4 attractors of length 1 (Figure 6). In the
Figures, nodes are labeled with organism state which is composed of the Boolean state of the cells.
Directed edges lead from an organism state to its successor state. Black edges connect tributaries
to their successor state and blue edges connect attractor states to their successor state [10].
3 Area 1: Cellular Differentiation and Robustness
3.1 Introduction
The evolutionary trend in organisms has been away from homogeneity and towards heterogeneity
and complex structures. Since natural selection is a driving force for evolution, there must be
some properties of the underlying networks [49] and the dynamics of those networks that are being
selected for [13] which drive evolution towards heterogeneity. Consequently, I propose the following
hypothesis: Cellular differentiation increases the expected robustness in an organism’s
dynamics.
When exploring the dynamics of an organism over infinite time, most of the time will be spend
cycling in attractors. As such, in this proposal the focus is only on attractor state perturbations
and the effects of those perturbations. Note, this assumption decreases the number of states whose
perturbations are computed and explored thereby also minimizing the computational overhead of
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Figure 4: Attractor length 4.
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Figure 5: Attractor length 2.
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Figure 6: Attractor length 1.8
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exploring both the state space and the perturbations.
3.2 Definitions
A mutation is the perturbation of a random cell state in the attractor state. In other words, it is a
bit flip of a single bit in the organism state [27]. The organism state that results from a mutation of
a single random cell state is a mutation state. Wuensche describes graphs of these perturbations
as attractor jump-graphs [57]. Mutation states are connected to the originating attractor state
by a mutation edge. Each organism state of an organism of size n has n possible resulting
mutations states and outgoing mutation edges. This type of perturbation is one of many that have
been explored. Perturbations can also be made in the cell function [54] resulting in a change in the
wiring or circuitry of the organism or in the timing or synchronicity of the signal [8]. There has
been research into the distance between cell states and cell functions, using the hamming distance
between two binary values [27][55]. For example, the distance D between a state 00000 and another
state 00111 is D = 3 since there are three bits with differing values.
A mutation state is an organism state that is either in an attractor or connected to an attractor
through directed edges in the dynamics graph. If that attractor or connected attractor is the
originating attractor then the mutation edge is called robust. Otherwise the mutation edge is not
robust. Robustness is the ability to resist external influences [6][11], and in the case of the organism
dynamics, robustness is the ability to conserve the dynamical topology [14][22][43]. Research where
the perturbation is in the synchronicity of signals applies the definition of robustness to be qualitative
changes resulting from perturbations in synchronicity in the update scheme [8].
Attractor robustness is computed by dividing the number of robust mutation edges by the
total number of mutation edges of the attractor states. An attractor of length L in the dynamics
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space of an organism of size n has n ⇤L mutation edges. Therefore, the robustness of that attractor
would be the number of robust mutation edges r/n ⇤ L.
Organism robustness ⇢ is the average attractor robustness of all the attractors in an organism’s
dynamics space. If the total number of attractors in the dynamics space of an organism is ↵, and
the robustness of attractor A is rA then
⇢ = (1/↵)
↵X
A=1
rA.
Note, that organism robustness could be computed as the average number of edges that are robust or
R/E where R is the number of robust mutation edges and E is the total number of mutation edges.
However, in this proposal, I apply the former definition such that organism robustness is the average
attractor robustness for all attractors in an organism’s dynamics space.
3.3 Formal Hypothesis 1
Applying the definitions, hypothesis 1 can be posed formally as follows. Let the average robustness
of the set of all heterogeneous organisms X at any fixed size N be Avg(⇢(X(N)))) and let the average
robustness of the set of all homogeneous organisms Y at the fixed size N be Avg(⇢(Y (N))) then for
N > 2
Avg(⇢(X(N))) > Avg(⇢(Y (N))).
3.4 Research Plan
The experimental methodology for testing this hypothesis will follow entail 3 components: program-
ming, simulation and data collection, and analysis.
The first stage is to develop a C program that sequentially computes the full organism state
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space, the number of attractors, and organism robustness. The program will also generate data files
of dynamics graph with mutation edges that can be rendered using graphviz [15]. There is existing
software, in particular Discrete Dynamics lab [58] is a tool that has been extensively used to model,
simulate, and study dynamical systems. Since the focus in this proposal is to explore properties that
differ from the properties studied before, the benefit of building my own code base is that it will be
highly customize-able. Developing a program instead of using the existing software further enables
optimization for larger organisms (exhaustive search of state space is limited, n 6 12 [55]) and to
develop novel algorithms for simulation (e.g. the later discussed simulation by multiple successor
steps).
The second stage is to simulate and collect data for analysis. The program will be run on organism
sizes that can be fully explored. The program will explore all 16 unique homogeneous organisms
for sizes n = 2 . . . 16. For organism sizes n = 2 . . . 16 the space of possible heterogeneous organisms
(n
16 � 16) is too large to fully explore. Therefore the program will also be run on a randomly
sampled set of 1000 heterogeneous organisms for sizes n = 2 . . . 16.
Finally, in the third stage data collected from the simulations will be analyzed at each size n
for homogeneous organism and for heterogeneous organisms to compare average robustness ⇢ and
average number of attractors ↵. Finally, the data will be used to compare dynamical properties
between the sets of homogeneous and heterogeneous organisms
3.5 Preliminary results
The number of attractors found from fully exploring all 16 homogeneous organisms of sizes = 2 . . . 16
can be divided into two classes: Class 1 organisms where the number of attractors remains bound
by some integer b as organism size increases (see figure 1), and Class 2 organisms where the
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Figure 7: Homogeneous organisms with bounded (left) and unbounded (right) numbers of attractors.
number of attractors grows unbounded (see figure 2). Given this preliminary result, the proposal
is to compare each of these classes of homogeneous organisms separately to the randomly sampled
heterogeneous organisms.
4 Area 2: Cellular Differentiation and Adaptivity
4.1 Introduction
Starting from a single cell, every organism is homogeneous. Therefore, this proposal seeks to identify
a property that when selected for would lead to the initial cellular differentiation from a homogeneous
organism to a "minimally" heterogeneous organism. In biological networks, specialization of cells
in heterogeneous organisms helps facilitate their adaptivity to environments. In an organism’s
dynamics the corollary to a group of specialized cells is an attractor. Consequently, the informally
proposed hypothesis is that cellular differentiation results in an increase in number of
attractors and in turn to an increase in adaptivity.
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Note that robustness and adaptivity are related properties both in nature and as dynamical
properties. Biologically, an organism can either be robust to a stimulus or adapt and evolve; as
such, there has been research into how an organism can be robust, while still evolving and adapting
[3]. In the previous area I propose exploring the dynamical property of robustness by varying the
organism property of homogeneity and heterogeneity while organism size is static. In this area, I
propose to focus on the dynamical property of adaptivity with respect to the organism property of
homogeneity and heterogeneity during growth.
Related work has involved researching spread [4] and influence [24] in the dynamics of social
networks. In particular the standing ovation model [38] and similar consensus problems are of
interest because the correlation in the dynamics of the organism is a collapse of adaptivity. However,
it is know that searching the dynamics space for specific attractor structures is computationally
challenging. Specifically, finding singleton attractors which have attractor length 1 is known to be
NP-hard [42][60][1].
4.2 Definitions
As mentioned in the preliminary definitions and introduction, there has been research into the ham-
ming distance between functions in an organism [27][55]. However, here I use the simple count of
different functions as a measure of homogeneity and heterogeneity of an organism. Therefore, a
homogeneous organism would have 0 different functions or formally |Im(f)| = 1, while a heteroge-
neous organism would have at least 1 different function or |Im(f)| > 1. Given these two distinct
classes of organisms, the border case is of interest. Specifically, an organism with 1 different func-
tion, formally |Im(f)| = 2, where every cell determines its state using the same function except for
one cell. I refer to this border case class of organism as minimal heterogeneous.
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Given the earlier definition of an attractor attractor, I can now define Adaptivity as the number
of attractors ↵(X) in the dynamics space of an organism. Attractor count is a measure of interest
that has been extensively studied [28][51][17]. I use the term adaptivity due to its relationship to
evolvability, defined by Aldana et al. to mean that an organism can acquire new funcitons and adapt
to new environments [3]. Further, if attractor count is considered memory [25], then organisms with
greater memory are more likely to be adaptive to new environments that require shifting memory.
4.3 Formal Hypothesis 2
Applying the definitions, hypothesis 2 can be posed formally as follows. Let the robustness of
a single homogeneous organism Y at size N be ⇢(YN ) and the robustness of a single minimally
heterogeneous organism X at size N be ⇢(XN ). Then as the organisms X and Y increase in size
from N = 2 . . .1,
9 i | i > 1 and ⇢(XN+i) > ⇢(YN+i).
4.4 Research Plan
The experimental methodology for testing this hypothesis will follow entail 3 components: program-
ming, simulation and data collection, and analysis.
The first stage will be to extend the C program from area 1 that sequentially explores the
organism state space to compute the number of attractors. The program will also generate data
files of dynamics graph with mutation edges that can be rendered using graphviz [15] and charted
using gnuplot [36]. In order to explore larger networks, I will expand the program to sampling the
state space and estimate attractor counts.
The second stage will simulate and collect data for analysis. The program will be run on organism
sizes that can be fully explored and sample the state space of for larger sizes. The program will
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0
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3
Figure 8: Homogeneous XOR size 2.
0
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Figure 9: Homogeneous XOR size 4.
Figure 10: Homogeneous XOR size 8.
explore the homogeneous XOR organisms for sizes n > 2 . . . 16. The program will also explore every
possible minimally heterogeneous organisms of the same size as the homogeneous XOR organisms
simulated.
In the third stage, analysis of the resulting number of attractors discovered for the homogeneous
XOR organism will be compared to the number of attractors found for each individual minimally
heterogeneous organism at every size organism whose dynamics space is fully explored and every at
every size organism whose dynamics space is sampled.
4.5 Preliminary results
The initial results of the dynamics space of homogeneous XOR organism exhibit a pattern of collapse
in adaptivity at sizes which are a power of 2 (see figures 8,9,10 and the chart in figure 11). Therefore,
for sizes n > 16 simulations will focus on organism sizes which are powers of 2.
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Figure 11: ↵ for homogeneous XOR organisms size N = 2 . . . 20.
5 Area 3: Symmetry and Growth
5.1 Introduction
In areas 2, adaptivity is explored with the assumption has been that organisms will grow incre-
mentally by a single cell from size n to n+1. However, growth rate is a network property that will
have its own relationship to dynamical properties. If growth rate impacts the dynamical properties
then natural selection would prefer a rate that increased or at least retained the desirable dynam-
ical properties. In area 2, the preliminary results show that for the homogeneous XOR organism
adaptivity is adversely impacted by incremental growth towards sizes that are powers of 2.
To introduce the idea of how organism growth could preserve symmetry, let’s look at the example
of a starfish. The starfish has clear rotational or radial symmetry. How can a starfish grow while
preserving rotational symmetry? There is the possibility to remain the same size while growing
another arm. And, there is the possibility to simply grow larger, where each part of the organism is
elongated. Both of these preserve its rotational symmetry. This proposal will look at both possible
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symmetry preserving patterns of growth.
In this area, I propose to explore how dynamical properties of symmetry can be preserved during
growth [5] in order to understand which patterns of growth might be selected for in organisms.
Previous work in symmetry of cellular automata has been explored by focusing on symmetry of
the functions [29][52], time symmetry referring to the successor function computing forwards and
backwards [16], and an organism’s dynamic state pattern symmetry resulting from an initial state
[48].
In area 1 and 2, I take a perspective of the organism as a whole and observe the global dynamics.
Since the focus in this area is on symmetry of the organism, in this area I will focus on the cell
dynamics as they relate to the global organism dynamics. As before in the assumption posited
in area 1, I will focus only on attractor states since from any intial state after an infinite number
of discrete time intervals, the organism will spend the majority of its time in attractor states.
Observing each individual cell of the organism as the organism orbits in an attractor, the individual
cells flip on and off in a period that divides the length of the attractor. Now, rather than the
instantaneous state of an organism in this area I consider the state of each cell over unbounded time
by examining the periodicity of its binary oscillations as the organism orbits in an attractor. The
periodic transition from 0 to 1 of the cell states may result in symmetric pattern of state changes
of cells in the organism cells during the orbit of an attractor. In this area, I propose to explore this
symmetry of cell periodicity in an attractor as a dynamical property in relation to organism growth
[10].
The focus in this area is symmetry. Here we will focus on rotational symmetry. Using the example
of a starfish that has clear rotational symmetry, I propose to look at two forms of growth which
could preserve that rotational symmetry. The first method of growth is cell by cell elongation of
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each "arm" simultaneously. The second method of growth is the reduplication or growth of an entire
new "arm" all at once. Taking inspiration from the biological organism, a starfish grows through
elongation rather then adding another arm: informally the proposed hypothesis is that growth by
elongation is more likely to preserve symmetry in dynamical properties then growth
by reduplication.
5.2 Definitions
Color is determined by the periodicity of a cell’s state as the organism traverses an attractor. Color
is assigned to an individual cell. Note that periodicity of cell state is dependent upon the attractor
being traversed; as such, for every attractor, each cell can have a different periodicity and coloring.
Because the periodicity has an upper bound of the length l of the attractor and a lower bound of 1
the possible colors for a cell will vary dependent upon the length of an attractor. As defined earlier,
the organisms are cyclic. Therefore, for each attractor in the dynamics of an organism, as each cell
is colored with respect to its periodicity, the organism itself takes on a pattern of corresponding
colors. In Figure 12 on the left, the homogeneous XOR organism of size 12 orbits an attractor of
length 4 and the. In Figure 12 in the middle, each of the 12 cells’ states are expanded outwards
towards infinity as the organism orbits in that attractor of length 4. In Figure 12 on the right, the
periodicity of the cells of the homogeneous XOR organism of size 12 can be seen as the organism
orbits an attractor of length 4, and corresponding colors are assigned to the organism.
The pattern of colors arranged in a ring can result in symmetry. In particular, the coloring could
be rotationally symmetry. Rotational symmetry in the case of coloring means that after a certain
amount of rotation the coloring would be the same. Segment size l is the number of cells which
when rotated result in the same coloring of an organism. Since rotation by segment size l results in
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the same coloring, segment size divides organism size. Foldedness k is the number of times segment
size divides into organism size k = N/l. Trivial rotational symmetry is defined as a segment size
l = 1 and foldedness k = n because it is the case where all cells have the same periodicity resulting
in an organism where all cells have the same color. This proposal is only interested in non-trivial
cases of symmetry.
In this area,Chromatic symmetry is defined to be non-trivial rotational symmetry in the
coloring of the organism cells as it traverses an attractor. A rotationally symmetric organism has
k segments of size l > 1 where each segment has the same pattern of coloring. In Figure 12 (on
the right), the coloring of the homogeneous XOR organism of size n = 12 as it orbits the attractor
of length 4 shown in Figure 12 (on the left) exhibits chromatic symmetry with segment size l = 6
and foldedness k = 2. This means that any the organism coloring can be rotated from any initial
position by a segment of 6 cells to the right or left and the resulting organism coloring will be the
same as the initial organism coloring [10].
4
1
244
42441
4
4
1
0
111
11100
1
0 1
0
001
00110
1
1
1
0
110
11110
0
10
0
011
10010
1
1
11101110...
10101010...
10111011...
11011101...
00000000...
11011101...
10111011...
10101010...
11101110...
01110111...
00000000...
01110111...
2381
3645869
3960
Figure 12: Attractor of size 4 in dynamics of homogeneous XOR organism size n=12.
Growth. Growth patterns have been extensively studied [2], for example for modeling morpho-
genesis in plants using L-systems [31]. Growth in areas 1 and 2 focused on incremental organism
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growth by a single cell from size n to size n+1. Reduplication is growth by incrementally increas-
ing the number of segments from k to k + 1. This increase in segments means that the organism
growth is from size n to size n+ l. Elongation is growth by incrementally increasing the segment
length from l to l+1. This increase in segment length means that every segment increases in length
by 1 and in turn the organism growth is from size n to size n + k. In Figure 13 the example of a
homogeneous XOR organism of size n = 10 in the orbit of an attractor of length 6 in its dynamics
graph Elongation gives a starting point from which the two forms of growth can be explored.
Note, that in the example the chromatic symmetry has segment size l = 5 and foldedness k = 2.
Figure 14 represents growth that occurs by segment elongation from a size n = 10 organism to a size
n = 12 organism, where the growth is an increased by increasing the segment size l = 5 to l = 6.
Figure 15 represents growth that occurs by segment reduplication from a size n = 10 organism to a
size n = 15 organism, where the growth is an increased by increasing the foldedness k = 2 to k = 3.
Note that in Figure 15 the segment size remains l = 5 [10].
5.3 Formal Hypothesis 3
Applying the definitions, hypothesis 3 can be posed formally as follows. Let A be a homogeneous
XOR organism of size N which has nontrivial rotational chromatic symmetry of segment length
LAi as the organism A traverses attractor i. Let Probability(sym(R)) be the probability that
there exists a homogeneous organism R of size N + LAi which has nontrivial rotational chromatic
symmetry of segment length LAi and let Probability(sym(E)) be the probability that there exists a
homogeneous organism E of size N + (N/LAi) which has nontrivial rotational chromatic symmetry
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0
0
000
00101
1
0
000
010001
1
010
01000
0
0
100
10000
0
1
010
00000
0
0
100
10101
6
6
661
66616
5
520
277
160
272
680
Figure 13: Coloring in orbit of attractor length 6 in dynamics of homogeneous XOR organism ofsize n=10
of segment length LAi + 1, then:
Probability(sym(E)) > Probability(sym(R))
5.4 Research Plan
The first stage will further extend the C program from area 2 to explore the dynamics of organisms
compute the chromatic symmetry of the organism across its attractors.
Because symmetry will be tested at larger organism sizes, the program will have to be able to
sample the dynamics space and handle larger numbers. Therefore, the program will be written
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4
4
414
44144
4
4
1
1
000
00000
0
0 0
0
101
00000
1
1
0
0
000
10001
0
00
0
000
01010
1
1
1345
3485
2176
Figure 14: Coloring in orbit of attractor length 4 in dynamics of homogeneous XOR organism ofsize n=12
3
13
33
31
33
13
33
33
1
10
10
01
01
00
00
01
0
10
11
01
11
00
10
10
1
11
00
11
00
01
01
01
8081
14453
11472
Figure 15: Coloring in orbit of attractor length 3 in dynamics of homogeneous XOR organism ofsize n=15
using the GNU MP big number library.
The second stage will simulate the organism dynamics space and collect data. The program will
explore the dynamics space of all homogeneous organisms and identify their possible colorings. Next,
selecting from the homogeneous organisms a subset that exhibit non-trival chromatic symmetry, the
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program will be run focusing on sizes which correspond to patters of growth that entail elongation
and reduplication.
In the third stage analyzing the data collected will require comparing the probability of preserving
symmetry by each growth pattern (elongation and re-duplication) in the sampled dynamics across
the simulated organism sizes.
5.5 Preliminary results
Attractor length bounds periodicity; therefore, organisms with attractor lengths of 1 are restricted to
the trivial case of chromatic symmetry. By sampling the dynamics of all the homogeneous organism
from 1000 random start states for sizes n = 2 . . . 50, preliminary data collection of attractor lengths
can be used to identify organisms whose coloring will be restricted to the trivial case of chromatic
symmetry. Figure 16 shows that only a subset of homogeneous organisms have increasing attractor
lengths and are therefore more likely to exhibit non-trivial chromatic symmetry.
1
10
100
1000
10000
100000
0 5 10 15 20 25 30 35 40 45 50
Attra
ctor
Len
gth
Organism Size
Attractor Lengths at Organism Size
All FunctionsFunction 6
Figure 16: Attractor lengths of homogeneous organisms of sizes N = 2 . . . 50
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The next step is to determine the relative frequency of the chromatic symmetry for these organ-
isms. The simulation and data collection will focus on only those homogeneous organisms that have
a non-zero relative frequency of chromatic symmetry.
6 Obstacles
6.1 Introduction
The experimental approach I propose taking faces obstacles that accompany big data. As the size
of a network grows linearly, the state space grows exponentially. Consequently, even at relatively
small network sizes, exhaustive search of the state space becomes computationally intractable. Be-
ing unable to exhaustively search the state space results in the inability to generate the complete
dynamics including identifying attractors, robustness, and chromatic symmetry. As the size of a
network grows linearly, the possible organisms grow exponentially. Further, the time and compu-
tational limitations of simulating each of these networks becomes computationally intractable even
at small network sizes.
In developing the programs, I propose the following three approaches to mitigating big data ob-
stacles: First, using algorithms which make use of patterns to increase efficiency in memory and
search by computing, storing, and comparing the successor state at multiplicative or exponential
time increments instead of single increments. Second, using sampling techniques to estimate prop-
erties of the dynamics and the organism. Third, using distributed processing to explore the state
space and sets of possible organisms in parallel
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6.2 Candidate Approaches
6.2.1 Powers of 2
Successor states are computed sequentially. Starting from a state, every next state is compared
against every previously viewed successor state of the initial state. Attractors are determined once
a previously seen state is seen again.
I propose to implement a more efficient algorithm then the trivial sequential successor computa-
tion with every state computed being compared with every previously computed successor state. If
P is the number of states from an initial state to the first state in an attractor + the length of the
attractor L, then the memory required is P states and the number of comparisons to discover an
attractor is (P 2)/2.
It is possible to lower the memory required for detecting an attractor by only storing a state if
it is at a successor step that is a power of 2. A side effect of the lower memory fewer results in
fewer comparison operations (log2p where p is the path explored so to the current successor state)
to check if the state has been visited before for each successor computation since there are fewer
states being stored. This algorithm requires P (log2P ) comparisons and log2 P memory.
For XOR organisms, I propose to work on developing an algorithm for simulating more then one
successor state at a time with a goal of computing forward in powers of 2 successor states at once.
The basis for this algorithm is the pattern observed in the preliminary results for area 2. The core
component of this algorithm relies on a proof that organisms of size n is a power of 2 in any state
will collapse to a single attractor after a set number C of successor computations. With a proof
of what the state of the organism will be after a set number of successor computations, then the
algorithm can be modified such that the incremental successor computations to operate in C steps
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instead of steps of size 1. Additional steps in the algorithm will be necessary to compute states that
need to be computed at intervals that differ from exact powers of C.
6.2.2 Sampling techniques
Random sampling is accomplished by picking a random state between 0 and (2N ) � 1, instead of
sequentially starting at each subsequent state in the state space and tracing it to an attractor.
Attractors are determined in the same way however, by computing the successor states from the
initial state until a previously seen state is seen again. For random sampling, I select a set D of
random states and trace them until I identify the attractor they fall into to.
For capture recapture, the program will randomly sample a number of states D, twice as 2
separate intervals. The set of attractors identified from the first set of D starting states are labeled
as Capture 1. The set of attractors identified from the second set of D starting states are labeled
as Capture 2. And, the set of attractors seen in both Capture 1 and Capture 2 are labeled as
Recapture.
Capture Recapture has limitations which occur as the size of the state space and the number of
attractors increase to such a degree that the sampled Capture 1 and Capture 2 have no overlap.
Further Capture Recapture assumes that the size and distribution of the basin of the attractors are
the same.
I propose developing a capture recapture technique which takes into account the varying attractor
sizes to estimate attractor counts of differing sized attractors and the total number of attractors in
an organisms dynamics space.
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6.2.3 Distributed processing
I propose to use a message passing interface to distribute sets of initial states to trace into attractors
to machines in a cluster. I will implement the distributed processing in the forensics lab cluster at
John Jay College of The City University of New York. Data computed by the simulation will then
be returned to a centralized server where it can be compressed and bulk inserted to a centralized
database that keeps track of the states explored. The bottleneck occurs at the centralized database.
Future work would be to decentralize the database to improve access time to the database for
reading and writing.
7 Summary
This proposal studies the impact of Darwinian selection on cellular differentiation and organism
growth. The proposal seeks a greater understanding of natural selection through exploring the re-
lationship of cellular differentiation to robustness and adaptivity and the relationship of growth to
chromatic symmetry. This proposal will follow an experimental approach to testing three hypothe-
ses:
• Cellular differentiation increases the expected robustness in an organism’s dynamics.
• Cellular differentiation leads to increasing adaptivity
• Organism growth by elongation preserves symmetry in the dynamics of an organism.
The proposed experimental approach faces obstacles related to big data. I propose three approaches
to addressing these obstacles: algorithms, sampling and distributed processing.
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8 Timeline
Date TaskOctober 2011 Research Plan 1November 2014 Research Plan 2November 2015 Research Plan 3May 2016 Dissertation ProposalJune-September 2016 Thesis writingSeptember 2016 Thesis defense
Table 4: Schedule of Tasks
Table 4 describes the time-line to complete the dissertation.
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