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\section*{Blog - evolution and categories}
This page is a [[blog article in progress]], written by [[Cameron Smith]]. To discuss this article while it's being written, visit the \href{http://www.math.ntnu.no/~stacey/Mathforge/Azimuth/comments.php?DiscussionID=773}{Azimuth Forum}.
An attempt to review some of the literature on \href{http://en.wikipedia.org/wiki/The_Major_Transitions_in_Evolution}{major transitions in evolution} and \href{http://en.wikipedia.org/wiki/Multi-level_selection#Multilevel_selection_theory}{multi-level selection}, sketch a few connections to concepts in category theory, and discuss the potential for using \href{http://en.wikipedia.org/wiki/Experimental_evolution}{experimental evolution} to investigate and strengthen those connections.
\href{http://www.azimuthproject.org/azimuth/edit/Blog+-+evolution+and+categories}{Edit}
\hypertarget{contents}{}\section*{{contents}}\label{contents}
\noindent\hyperlink{simple_and_complex_systems}{simple and complex systems}\dotfill \pageref*{simple_and_complex_systems} \linebreak
\noindent\hyperlink{nested_and_nonnested_hierarchies}{nested and non-nested hierarchies}\dotfill \pageref*{nested_and_nonnested_hierarchies} \linebreak
\noindent\hyperlink{the_organismenvironment_duality}{the organism-environment duality}\dotfill \pageref*{the_organismenvironment_duality} \linebreak
\noindent\hyperlink{multilevel_selection}{multi-level selection}\dotfill \pageref*{multilevel_selection} \linebreak
\noindent\hyperlink{major_transitions_in_evolution}{major transitions in evolution}\dotfill \pageref*{major_transitions_in_evolution} \linebreak
\noindent\hyperlink{a_categorical_view_of_evolutionary_processes}{a categorical view of evolutionary processes}\dotfill \pageref*{a_categorical_view_of_evolutionary_processes} \linebreak
\noindent\hyperlink{references}{references}\dotfill \pageref*{references} \linebreak
\begin{quote}%
My thesis has been that one path to the construction of a non-trivial theory of complex systems is by way of a theory of hierarchy. Empirically, a large proportion of the complex systems we observe in nature exhibit hierarchic structure. On theoretical grounds we could expect complex systems to be hierarchies in a world in which complexity had to evolve from simplicity. -\href{http://en.wikipedia.org/wiki/Herbert_Simon}{Herbert Simon}, 1962
\end{quote}
The prospects of a general theory of evolution point to an intellectual challenge inherent to the interrelationships between theories relevant to the domain of empirical science. Consider the \emph{level of organization} concept. The presumption of energetic stratification among the contents of the universe has led to a corresponding stratification among the scientific theories that intend to address each of the perceived layered levels (\hyperlink{Farre1997}{Farre, 1997}). These layers have come to be conceived of as levels of organization and scientific theories tend to address one of these levels (click the image to see the flash animation):
It might be useful to work explicitly on connecting theories that transcend levels of organization. One type of insight that could be gained from this approach is an understanding of the mutual development of bottom-up \emph{ostensibly mechanistic} models of simple systems and top-down \emph{initially phenomenological} models of complex ones.
Taking a \href{http://plato.stanford.edu/entries/scientific-unity/}{monistic ideological perspective of science}, a significant challenge is the development of a language able to unify descriptions of both simple and complex systems (see \hyperlink{Simon1962}{Simon, 1962} for a discussion of the quasi-continuum that ranges from simple to complex). Such a language is essential to facilitate efficient communication among scientists who work with complex systems that apparently transcend multiple levels of organization. Category theory may provide the nucleus of the framework necessary to formally address this challenge. But, in order to head in that direction, I'll try out a few examples, albeit from the somewhat limited perspective of a biologist, from which some pattern(s) might begin to surface.
\hypertarget{simple_and_complex_systems}{}\subsection*{{simple and complex systems}}\label{simple_and_complex_systems}
[[Blog - evolution and categories]]
What is the organizational structure of the products of evolutionary processes? Herbert Simon provides one perspective that I find intuitive in his \href{http://www.mech.kuleuven.be/goa/hms-int/parable.html}{parable of two watchmakers} (\hyperlink{Simon1962}{Simon, 1962}).
He argues that the existence of multiple stable intermediate modules that can be assembled hierarchically reduces the magnitude of the \emph{timescale}, relative to that for systems lacking stable intermediates, for the evolution of complex systems. Given a particular set of internal and environmental constraints that can only be satisfied by some relatively complex system, a hierarchically organized one will be capable of meeting those constraints with the fewest resources and in the least time (i.e. most efficiently). Constraints inherent to a process define ``stability'', and thus dictate the types of structures permitted to exist in its context. If \emph{hierarchical} organization is an unavoidable outcome of evolutionary processes, it should be possible to characterize the \emph{generators} that lead to its emergence.
Simon describes a property that most complex systems have in common that he refers to as \emph{near decomposability} (ND) (\hyperlink{Simon2002}{Simon, 2002}). Simon uses a coefficient matrix to describe a ND system:
\begin{quote}%
This matrix illustrates a ND system of three layers, with two cells at the top level, each divided into two subcells, and each of these subdivided into three sub-subcells. The numbers of the rows and columns designate the cells, cells 1–6 and 7–12 constitute the two top-level subsystems, cells 1–3, 4–6, 7–9 and 10–12 the four second-level sub- systems. The interactions within the latter subsystems have intensity $a$, those within the former two subsystems, intensity $\epsilon_l$, and those between components of the largest subsystems, intensity $\epsilon_2$. (\hyperlink{Simon2002}{Simon, 2002})
\end{quote}
Simon states that this matrix is in \emph{near-diagonal form} because he assumes $a\gg\epsilon_l\gg\epsilon_2$. Another, probably more common, terminology for this would be \emph{near} \href{http://en.wikipedia.org/wiki/Block_matrix#Block_diagonal_matrices}{block diagonal}. Simon also tells us that assuming linearity, this system can be modeled by a system of differential equations:
\begin{displaymath}
\frac{d}{d t}T_i(t) = \sum_{j}a_{ij}\left(T_{j}(t)-T_{i}(t)\right),
\end{displaymath}
where $i$ indexes rows, $j$ indexes columns, $a_ij$ correspond to the entries (i.e. $a,\,\,\epsilon_l,\,\,\epsilon_2$) in the coefficient matrix, and the variables $T_i$ (probably $T$ because he's modeling the temperature in a fictional building called the Mellon institute) represent the variables at the lowest level of the hierarchy (in this case one of the twelve).
This is a trivial system, but it illustrates that the ND of the coefficient matrix allows these equations to be solved in a \emph{near} hierarchical fashion. As an approximation, rather than simulating all the equations (e.g. twelve in this example but more generally $\forall\,i \in 1...m$) one can take a recursive approach and solve the four systems of three equations (each of the blocks containing $a$s), and average the results to produce initial conditions for two systems of two equations with coefficients:
\begin{displaymath}
\begin{bmatrix}
\epsilon_1 & \epsilon_1 & \epsilon_2 & \epsilon_2 \\
\epsilon_1 & \epsilon_1 & \epsilon_2 & \epsilon_2\\
\epsilon_2 & \epsilon_2 & \epsilon_1 & \epsilon_1 \\
\epsilon_2 & \epsilon_2 & \epsilon_1 & \epsilon_1
\end{bmatrix},
\end{displaymath}
and then average those results to produce initial conditions for a single system of two equations with coefficients:
\begin{displaymath}
\begin{bmatrix}
\epsilon_2 & \epsilon_2 \\
\epsilon_2 & \epsilon_2
\end{bmatrix}.
\end{displaymath}
This example of heuristic simplification indicates that for systems possessing the ND property, a single system could be reduced to a series of smaller modules, which can be simulated in less computational time, if the error introduced in this approximation is tolerable. The degree to which this method saves time is dependent upon the relationship between the size of the whole system and the size and number of hierarchical levels. However, as an example, given the \href{http://en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations#Matrix_algebra}{time complexity for matrix inversion} (i.e. solving a system of linear equations) is $O(n^2)$ then the hierarchical decomposition would lead to an algorithm with time complexity $O\left(\left(\frac{n}{L}\right)^2\right)$, where $L$ is the number of levels in the decomposition ($L=4$ in Simon's example assuming the individual variables represent the lowest level).
All of this will need to be made much more precise; however, there are some potential metaphorical consequences for the evolution of complex systems:
\begin{quote}%
If we begin with a population of systems of comparable complexity, some of which are ND and some of which are not, the ND systems will, on average, increase their fitness through evolutionary processes much faster than the remaining systems, and will soon come to dominate the entire population. Notice that the claim is not that more complex systems will evolve more rapidly than less complex systems, but that, at any level of complexity, ND systems will evolve much faster than systems of comparable complexity that are not ND. (\hyperlink{Simon2002}{Simon, 2002})
\end{quote}
\hypertarget{nested_and_nonnested_hierarchies}{}\subsection*{{nested and non-nested hierarchies}}\label{nested_and_nonnested_hierarchies}
[[Blog - evolution and categories]]
Even \emph{within} the molecular level of biological systems, we observe a form of structural:
,
and functional:
,
hierarchical organization. The first picture shows the physical structure of the genome. The structural hierarchy here from micro$\rightarrow$macro is roughly DNA$\rightarrow$Nucleosome$\rightarrow$Chromatosome$\rightarrow$ fibers of increasing size\ldots{} (see the descriptions in the numbered bubbles). The second figure shows several of the components involved in the regulation of gene expression. A protein called a transcriptional activator (there are also repressors) may bind to a particular type of DNA sequence called a promoter and recruit another type of protein called RNA polymerase to promote (i.e. in a stochastic model, increase the probability of\ldots{}) the production of complementary RNA from the DNA template. However, the transcriptional activator may be prevented from accessing the promoter DNA if the DNA is tightly wound up around the proteins called histones. The enzymes that regulate the degree to which certain stretches of the DNA are \emph{condensed} around histones are called histone modification enzymes. The process by which histone modification enzymes regulate gene expression is a form of \emph{\href{http://en.wikipedia.org/wiki/Epigenetics}{epigenetic regulation}} (of which there are several other types). Thus, we see that here we have something that might be termed a \emph{functional} hierarchy wherein the histone modification enzymes represent a meta-level control of gene expression with respect to that of transcription factors (TFs being a slightly more general term subsuming both activators and repressors).
There is a property that allegedly differs between these examples of one structural and one functional hierarchy. The structural hierarchy is \emph{\href{http://www.isss.org/hierarchy.htm}{nested}} (all lower levels are physically contained within all higher levels) whereas the functional one is not (\hyperlink{AllenStarr1982}{Allen and Starr, 1982}).
Presumably, the structure of the genome is itself functional, and the way in which I have presented genome organization and regulation has implied a false dichotomy. I believe the description I have provided above is representative of what most young biologists are taught. In the light of evolution, multi-level selection theory in particular, the genome structure can be viewed as a functional hierarchy whose timescale of change is, on average, longer than that of the expression of a gene. The tendency to impose a semantic distinction between ``structural'' and ``functional'' hierarchies, as I have recapitulated here, is, I suspect, simply a method of implicitly expressing relative timescales of stability. Nested hierarchies may evolve from non-nested hierarchies if the relatively transient interactions that constitute the non-nested hierarchy are stabilized via selection. If this occurs, however, a previously non-existant \emph{level} emerges at which a non-nested hierarchy may come to exist. If this process were iterated, nested hierarchies emerging from non-nested ones and vice versa, it would bring into existence succesively higher levels: non-nested$_n$ $\stackrel{n=n+1}{\leftrightarrow}$ nested$_n$.
\hypertarget{the_organismenvironment_duality}{}\subsection*{{the organism-environment duality}}\label{the_organismenvironment_duality}
[[Blog - evolution and categories]]
Population genetics is a branch of biology that takes as its mode of abstraction the proportions of particular genotypes within a given population. A meaningful conceptualization of \href{http://en.wikipedia.org/wiki/Fitness_(biology}{fitness}, for which concrete examples can be devised, seems to me to be a significant impediment to population genetics theory. One reason for this is that any type of \emph{individual entity}, no matter the hierarchical level on which it is perceived to exist, is composed of a network of traits that interact both among themselves (e.g. \href{http://en.wikipedia.org/wiki/Epistasis}{epistasis} among genes) and with a dual network of factors that constitute said individual entity's environment. It might be useful to go further and abstract away the conceptual division between organism and environment altogether, in order to incorporate both into a single network, as I believe many theoretical ecologists would argue. But, if we are to maintain the conceptual organism-environment distinction, I think it will be necessary in constructing a higher-level model to incorporate models of the environment which are at least equally complex to that of the organism. As an example in this direction, the notion of a so-called \href{http://en.wikipedia.org/wiki/Fitness_landscape}{fitness landscape} as exhibited by the \href{http://en.wikipedia.org/wiki/NK_model}{NK model} takes a step in the right direction. The relationship between organism and environment is more like \href{http://en.wikipedia.org/wiki/Geometrical_frustration}{frustrated} \href{http://en.wikipedia.org/wiki/Spin_glasses}{spin glasses} than a smooth climb up a \href{http://en.wikipedia.org/wiki/Fitness_landscape}{single global fitness peak}. Probably the closest concept to frustration that has been directly studied in biology is that of \href{http://en.wikipedia.org/wiki/Host-parasite_coevolution}{host-parasite antagonistic coevolution}, which I would use as a metaphor to \emph{environment-organism antagonistic coevolution} with environment and organism being analogous to host and parasite respectively. One naïve approach is to study population genetic models, which tend to take the organism as their focal point, and just upon reaching what seems to be a satisfactory description of an organism, or population thereof, turn the conceptualization on its head to determine what has been implied about the environment. It may then be possible to enrich the environmental description to be on par with that of the biological entities of focus (e.g. molecules, organisms, populations, \ldots{}).
\hypertarget{multilevel_selection}{}\subsection*{{multi-level selection}}\label{multilevel_selection}
[[Blog - evolution and categories]]
\begin{quote}%
A model that unifies all types of selection (chemical, sociological, genetical, and every other kind of selection) may open the way to develop a general `Mathematical Theory of Selection' analogous to communication theory. -\href{http://en.wikipedia.org/wiki/George_R._Price}{George R. Price}, 1971
\end{quote}
The \href{http://en.wikipedia.org/wiki/Price_equation}{Price equation} (sometimes referred to as Price's theorem) provides a statistical description of an evolutionary process. It partitions the average change in the value of the population averaged value of a \emph{trait} $z\,\,(\Delta z)$ between generations into components due to selection and transmission. What is meant by selection and transmission can begin to be understood from a cartoon:
,
\begin{quote}%
Example of a selective system using the notation of the Price Equation. The initial population, the left column of beakers, is divided into subpopulations indexed by $i$, where $q_i$ is the fraction of the total population in the $i$-th subpopulation. In this drawing, the two different kinds of transmissible material, solid and striped, are in separate subpopulations initially, but that is not necessary. Each subpopulation expresses a character value (phenotype), $z_i$. Any arbitrary rule can be used to assign trait values. Selection describes the changes in the quantities of the transmissible materials, where the primes on symbols denote the next time period. Thus $q'_i = q_i w_i/\bar{w}$ is the proportion of the descendant population derived from the $i$-th subpopulation of the initial population. The transmissible material may be redistributed to new groupings during or after the selective processes. The $q'_{j \dot i}$ are the fractions of the $i$-th parental subpopulation, after selection, that end up in the $j$-th descendant subpopulation, thus $\sum_j q'_{j \dot i} = 1$. The new mixtures in the $j$-th subpopulations express trait values $y_j$ according to whatever arbitrary rules are in effect. This allows full context-dependence (non-additivity) in the phenotypic expression of the transmissible material. Descendant trait values are assigned to the original subpopulations by weighting the contributions of those subpopulations, $z'_i = \sum_j q'_{j \dot i} y_j$. Thus, the average trait value in the descendant population is $\bar{z}' = \sum_i q'_i z'_i$ (\hyperlink{Frank1995}{Frank, 1995}, Fig. 1).
\end{quote}
The two terms in the Price equation are:
\begin{itemize}%
\item the covariation between the fitness relative to the population average $\left(v_i=\frac{w_i}{w}\right)$ and the trait value,
\item and the fitness-weighted expected value of the change in the trait value between generations (\hyperlink{Price1970}{Price, 1970}, \hyperlink{Price1972}{Price, 1972}, \hyperlink{Price1995}{Price, 1995}).
\end{itemize}
First I'll provide a construction of the Price equation, using slightly different notation from the figure legend above, and then discuss it's meaning in more detail.
\begin{udefn}
If $n_i \in \mathbb{Z}^*$ is the number of occurrences for each $x_i,\,y_i \in \mathbb{R}$ then:
The \emph{expected value} of the $x_i$ values weighted by $n_i$ is:
\begin{displaymath}
\operatorname{E}(x_i) \stackrel{\mathrm{def}}{=} \frac{\sum_i x_i n_i}{\sum_i n_i}.
\end{displaymath}
The \emph{covariance} between the $x_i$ and $y_i$ values weighted by $n_i$ is:
\begin{displaymath}
\operatorname{Cov}(x_i,y_i) \stackrel{\mathrm{def}}{=} \frac{\sum_i n_i[x_i-\operatorname{E}(x_i)][y_i-\operatorname{E}(y_i)]}{\sum_i
n_i}
= \operatorname{E}(x_i y_i)-\operatorname{E}(x_i)\operatorname{E}(y_i).
\end{displaymath}
\end{udefn}
\begin{udefn}
Suppose there is a population (a set) wherein each individual entity has a characteristic described by a number$\in \mathbb{R}$. For example, high values of the number for one individual represent an increased value for that characteristic over some other individual with a lower value. \emph{Value} is not equivocated here with \emph{fitness} (to be defined). Let subscript $i$ identify the group with characteristic values $z_i$ and let $n_i$ be the number of individuals in that group. The \emph{total number of individuals} is then $n$ where:
\begin{displaymath}
n = \sum_i n_i.
\end{displaymath}
The \emph{average value of the characteristic}, $z$, is defined as:
\begin{displaymath}
z \stackrel{\mathrm{def}}{=} \operatorname{E}(z_i) = \frac{1}{n} \sum_i z_i n_i.
\end{displaymath}
\end{udefn}
\begin{udefn}
Suppose that the population reproduces, all parents are eliminated, and there is a selection process on the offspring, by which offspring deemed less fit are removed from the reproducing population. After reproduction and selection, the population numbers for the offspring groups will change to $n'_j$. Primes or $j-indices$ denote relevance to the offspring population, and the absence of primes or $i-indices$, the like for the parent population.
The \emph{total number of offspring} is $n'$ where:
\begin{displaymath}
n' = \sum_i n'_i.
\end{displaymath}
\end{udefn}
\begin{udefn}
The \emph{fitness} of group $i$ will be defined to be the within-group ratio of offspring to parents:
\begin{displaymath}
w_i = \frac{n'_i}{n_i},
\end{displaymath}
with average fitness of the population being
\begin{displaymath}
w \stackrel{\mathrm{def}}{=} \operatorname{E}(w_i) = \frac{1}{n} \sum_i w_i n_i = \frac{1}{n} \sum_i \frac{n'_i}{n_i} n_i = \frac{1}{n} \sum_i n'_i = \frac{n'}{n}.
\end{displaymath}
The average value of the offspring characteristic will be $z'$ where:
\begin{displaymath}
\begin{array}{rcl}
z' & = & \frac{1}{n'} \sum_{j} z'_j n'_j, \\
& = & \frac{1}{n} \sum_{i} \frac{w_i}{w} z'_i n_i,
\end{array}
\end{displaymath}
$z'_i$ represent the \emph{average} character values of the offspring of each group $i$, and $z'_j$ represent the character values of the offspring for the (potentially different) groupings $j$. Thus, the average character value of the offspring population, $z'$, can be determined summing over the parent, $i$, or offspring, $j$, population groupings.
\end{udefn}
\begin{theorem}
\label{Price}\hypertarget{Price}{}
\textbf{(Price's theorem)}. The change in the population averaged trait value, $\Delta z = z'-z$, between generations is given by:
\begin{displaymath}
w\,\Delta z = \operatorname{Cov}(w_i,z_i)+\operatorname{E}(w_i\,\Delta z_i)
\end{displaymath}
\end{theorem}
\begin{proof}
Equation (2) shows that:
\begin{displaymath}
\operatorname{Cov}(w_i,z_i)=\operatorname{E}(w_i z_i)-w z
\end{displaymath}
Call the change in characteristic value from parent to child populations $\Delta z_i$ so that $\Delta z_i = z'_i - z_i$. As seen in Equation (1), the expected value operator $\operatorname{E}$ is linear, so
\begin{displaymath}
\operatorname{E}(w_i\,\Delta z_i)=\operatorname{E}(w_i z'_i)-\operatorname{E}(w_i z_i)
\end{displaymath}
Combining Equations (7) and (8) leads to
\begin{displaymath}
\operatorname{Cov}(w_i,z_i)+\operatorname{E}(w_i\,\Delta z_i)
= \bigl(\operatorname{E}(w_i z_i)-w z \bigr) + \bigl(\operatorname{E}(w_i z'_i)-\operatorname{E}(w_i z_i)\bigr)
= \operatorname{E}(w_i z'_i)-w z
\end{displaymath}
but from Equation (1) gives:
\begin{displaymath}
\operatorname{E}(w_i z'_i)=\frac{1}{n} \sum_i w_i z'_i n_i
\end{displaymath}
and from Equation (4) gives:
\begin{displaymath}
\operatorname{E}(w_i z'_i)=\frac{1}{n} \sum_i\frac{n'_i}{n_i}z'_i n_i = \frac{1}{n} \sum_i n'_i z'_i=\frac{n'}{n}\frac{\sum_i z'_i n'_i}{n'}
\end{displaymath}
Applying Equations (5) and (6) to Equation (10) and then applying the result to Equation (9) gives the Price Equation:
\begin{displaymath}
\operatorname{Cov}(w_i,z_i)+\operatorname{E}(w_i\,\Delta z_i)=w z'-w z=w\,\Delta z
\end{displaymath}
\end{proof}
\begin{udefn}
Let the \emph{relative fitness} of individual $i$ be:
\begin{displaymath}
v_i \stackrel{\mathrm{def}}{=} \frac{w_i}{w}
\end{displaymath}
\end{udefn}
\begin{lemma}
\label{RelFitPrice}\hypertarget{RelFitPrice}{}
The \emph{relative fitness} formulation of the Price equation is:
\begin{displaymath}
\Delta z = \operatorname{Cov}(v_i,z_i)+\operatorname{E}(v_i\,\Delta z_i)
\end{displaymath}
\end{lemma}
\begin{proof}
Combining Equation (12) with Theorem \ref{Price} and the definitions of the $\operatorname{E}$ and $\operatorname{Cov}$ operators in Equations (1) and (2) respectively:
\begin{displaymath}
\begin{array}{rcl}
\Delta z & = & \operatorname{Cov}\left(\frac{w_i}{w},z_i\right)+\operatorname{E}\left(\frac{w_i}{w}\,\Delta z_i\right) \\
& = & \operatorname{Cov}(v_i,z_i)+\operatorname{E}(v_i\,\Delta z_i)
\end{array}
\end{displaymath}
\end{proof}
It is essential to emphasize the change of indices between $j$ (offspring) and $i$ (parents) in Equation (6). This indicates that the descendant population can have different groupings from the parent population. If we actually want to write down an explicit method of computing $z'_i$, we have to have kept track of something about the relationship between the groupings in the parent and child populations:
\begin{displaymath}
z'_i = \frac{1}{n_i} \sum_j n_{i j} z'_j,
\end{displaymath}
where $n_{i j}$ represents the number of individuals in group $j$ of the offspring population derived from group $i$ of the parent population (see the figure legend above and (\hyperlink{Frank1995}{Frank, 1995})). This seems to be a significant limitation because it requires interrogation of the distribution of the offspring population. In fact, it also implies that the price equation is purely retrospective rather than predictive.
What is the meaning of the Price equation? The first term, $\operatorname{Cov}(v_i,z_i)$, measures the statistical relationship between the trait value, $z$, and the relative fitness $v$. That is, if higher values for a particular trait are positively correlated with relative fitness this term will be positive and if negatively correlated it will be negative. If there is no statistical association or if there is variation neither in relative fitness nor trait value, then $\operatorname{Cov}(v_i,z_i)=0$. Thus, $\operatorname{Cov}(v_i,z_i)$ measures the degree to which the trait in question is subject to selection, thereby encapsulating the organism-environment relationship \emph{for a single trait}. It is sometimes called the \href{http://en.wikipedia.org/wiki/Heritability#Selection_experiments}{selection differential}. The second term in the Price equation, $\operatorname{E}(v_i\,\Delta z_i)$, is a measure of the so-called \emph{transmission bias} of the trait. Each individual in the population has a $\Delta z_i$, so $\operatorname{E}(v_i\,\Delta z_i)$ is a fitness-weighted expectation that relays both the degree to which the offspring of individual $i$ deviate from it in trait value as well as its number of offspring.
What is perhaps most interesting about the Price equation is that it can be expanded into what might be conceptualized as an arbitrary number of hierarchical levels due to its recursive nature. To my knowledge, Arnold and Fristrup were the first to describe this in detail (\hyperlink{ArnoldFristrup1982}{Arnold and Fristrup, 1982}). The following is a construction of the hierarchical formulation of the Price equation for an arbitrary number of hierarchical levels.
\begin{udefn}
Given an ordered set of sets of indices $\mathfrak{I} = \{ \mathbb{K}, \mathbb{L}, \ldots\, , \mathbb{M} \}$ where $\mathbb{K}= \{ k | k \in \mathbb{Z}^*, k = 1 \ldots\, K \}$ there are operations on the index sets such that $\mathbb{K} \oplus 1 = \mathbb{L}$ and $\mathbb{L} \ominus 1 = \mathbb{K}$.
\end{udefn}
\begin{lemma}
\label{HigherPrice}\hypertarget{HigherPrice}{}
The nested hierarchical form of the Price equation is:
\begin{displaymath}
\begin{split}
w\,\Delta z =& \operatorname{Cov}_k(w_k,z_k) \\
& +\, \operatorname{E}_k ( \operatorname{Cov}_l(w_kl,z_kl) \\
& +\, \operatorname{E}_l (\,\, \cdots\,\, \operatorname{E}_{m \ominus 1}( \operatorname{Cov}_m(w_{k l\, \cdots\, m},z_{k l\, \cdots\, m}) \\
& +\, \operatorname{E}_m(w_{k l\, \cdots\, m} \Delta z_{k l\, \cdots\, m}))\,\, \cdots \,\,)).
\end{split}
\end{displaymath}
The number of levels in the hierarchy is equal to $|\mathfrak{I}|$.
\end{lemma}
\begin{proof}
Equation (15) is a direct result of writing the terms describing $w\,\Delta z$ in succession:
\begin{displaymath}
\begin{array}{rcl}
w\,\Delta z &=& \operatorname{Cov}(w_k,z_k)+\operatorname{E}(w_k\,\Delta z_k), \\
w_k\,\Delta z_k &=& \operatorname{Cov}_l(w_{k l},z_{k l})+\operatorname{E}_l(w_{k l}\,\Delta z_{k l}), \\
w_{k l}\,\Delta z_{k l} &=& \operatorname{Cov}_{l \oplus 1}(w_{k l (l \oplus 1)},z_{k l (l \oplus 1)})+\operatorname{E}_{l \oplus 1}(w_{k l (l \oplus 1)}\,\Delta z_{k l (l \oplus 1)}), \\
&\vdots& \\
w_{k l \cdots\, m \ominus 1}\,\Delta z_{k l \cdots\, m \ominus 1} &=& \operatorname{Cov}_m(w_{k l \cdots\, m},z_{k l \cdots\, m}) + \operatorname{E}_m(w_{k l \cdots\, m} \Delta z_{k l \cdots\, m}).
\end{array}
\end{displaymath}
Equation (15) results from substitution of each of Equations (16) into its immediate predecessor until the first of the set of equations (16) is reached.
\end{proof}
The meaning of the \emph{hierarchical formulation} of the Price equation can be understood by considering each of Equations (16) as encapsulating the separation of the evolutionary process of some abstract characteristic into selection and transmission components \emph{each} for a particular \emph{hierarchical level}. Equation (15) is simply the composition of all of Equations (16). In their current form Equations (16) cannot be viewed as a mapping with a domain and co-domain. If we were to try to use the difference Equations (16) to update the necessary input data over a series of iterations, then defining the domain appears to require knowledge of characteristic values decomposed into the relevant set of hierarchical levels and summarized in some tensor $\mathbf{Z}_{K L\, \cdots\, M}$ with $\operatorname{dim} \{ \mathbf{Z} \} = K \times L \times \cdots\, \times M$. A similar tensor is required to describe the fitness values $\{w_{k l\, \cdots\, \m} \}$. In each case, however, we need equivalent tensors for the offspring population in order to compute the $z'_i$. In order to update the characteristic value tensor for a single iteration we would have to add an average over a dimension to each indexed element in that dimension. Thus, information required to fully reconstruct rather than summarize statistically the character value distribution of the offspring of each parent or parent group would be lost in each generation. In summary, as mentioned above, despite its potential usefulness as a conceptual tool, in its current form we could only use the Price equation to compare to an experiment if we had complete data for both the parent and offspring populations over all generations of interest. Later I'll attempt to derive a stochastic version of the Price equation that might be more useful in attempting to embed it, or something else, within a higher-level framework.
I'll probably need a lot of help with this over on the forum!
Another limitation of the Price equation is that the fitness values do not evolve themselves. Referring back to the \href{levels+of+organization#the_organismenvironment_duality_4}{organism-environment duality} it appears here that, while organisms in the environment are dynamic, the environment, which ultimately \emph{defines} fitness is static. While this may be true over appropriately defined timescales, it is certainly not true in general. A model taking into account this intuitive relationship, while refusing to let go of a distinction between organisms and environments, would allow for the coevolution of the environment (thus fitness). Alternatively, we could take the point of view that an organism, indeed all organisms, simply represent a particular subset of hierarchical levels embedded within a larger network representing the ``environment''. In this context, the underlying levels, molecules, organisms, populations, communities, ecosystems, and the overlying levels may all be subsumed into a single framework wherein something like an ``organism'' would simply represent a pattern that can be detected via some, ideally analytical, means within a single ``global'' type of network. This is perhaps where the project I'm working on converges in some sense with that of \href{http://math.ucr.edu/home/baez/networks/networks.html}{John and Jacob}.
\hypertarget{major_transitions_in_evolution}{}\subsection*{{major transitions in evolution}}\label{major_transitions_in_evolution}
[[Blog - evolution and categories]]
One of the first attempts to consider so-called \href{http://en.wikipedia.org/wiki/The_Major_Transitions_in_Evolution}{major transitions in evolution} \emph{comprehensively} was Maynard Smith and Szathmary's (\hyperlink{SmithSzathmary1995}{Maynard Smith and Szathmary, 1995})
Still putting together what I want to say about this\ldots{}
\hypertarget{a_categorical_view_of_evolutionary_processes}{}\subsection*{{a categorical view of evolutionary processes}}\label{a_categorical_view_of_evolutionary_processes}
[[Blog - evolution and categories]]
This is clearly the least developed and probably most important section with which I will require the most help if it is to become comprehensible or useful!: \href{http://www.math.ntnu.no/~stacey/Mathforge/Azimuth/comments.php?DiscussionID=773}{Azimuth Forum}.
Among the capabilities of the categorical language applied to the science of complex systems as described in \hyperlink{Ehresmann2007}{Ehresmann and Vanbremeersch} are, apparently, a characterization of the interface between simple and complex, a statement of the necessary conditions for reductionist theories to be successful in recapitulating natural systems, and a corresponding identification of the range of considerations necessary to construct reliable models when reductionism is insufficient. All of these constructions discussed in the first several chapters of MES are intellectually tantalizing, and it will be interesting to see where all of the directions suggested in MES could lead us. In any case, what I hope to discuss here, is a bit more modest.
Although recklessly premature, I cannot resist juxtaposing these two figures:
\vspace{.5em} \hrule \vspace{.5em}
There is probably some other structure (or at the very least a series of adjectives may need to be added for any modicum of precision) more appropriate for embedding either a model of hierarchical evolution, or the hierarchical price equation$-$which are probably not the same given that the former expresses an ideal and the latter a practical result in the potential direction of that ideal. The rough notion of an evolutionary process modeled as a \href{http://ncatlab.org/nlab/show/fibration#fibrations_in_category_theory_5}{fibration} wherein there is a, perhaps $n-$, category representing a collection of interacting objects that constitute an evolving system whose slicing parameter is identified with time. This is how EV define it:
\begin{udefn}
If $\mathbf{K}$ is an evolutive system, the fibration $\mathbf{FK}$ associated to $\mathbf{K}$ is a quasi-category which has for its set of objects the set $|K|$ of all the objects of $\mathbf{K}$, and which is generated by its following sub-categories (see the figure above):
\begin{enumerate}%
\item The configuration categories $K_t$ for each $t$; their links are called \emph{vertical links}. $K_t$ is also called the fiber at $t$.
\item The category associated to the order `earlier than or simultaneous with' on $|K|$; these links are called \emph{horizontal links}.
\end{enumerate}
\end{udefn}
There are many definitions before and after this one in MES; however, the meaning of EV's term \emph{evolutive system} is probably the least decipherable without a slightly more explicit definition:
\begin{udefn}
An evolutive system (or $ES$) $\mathbf{K}$ consists of the following (see figure below):
\begin{enumerate}%
\item A time scale $T$, which is an interval or a finite subset of $\mathbb{Z}^*$.
\item For each instant $t$ of $T$, a category $K_t$ called the \emph{configuration category} at $t$. These categories are disjoint.
\item For each instant $t' \gt t$, a partial functor $k(t,t')$ from $K_t$ to $K_{t'}$, called the transition from $t$ to $t'$. These transitions satisfy the following transitivity condition (TC), given $t \lt t' \lt t''$ in $T$:
\end{enumerate}
: (TC) If the object $A_t$ has $A_{t'}$ for its new configuration at $t'$, and if $A_{t'}$ has a new configuration $A_{t''}$ at $t''$, then $A_{t''}$ is also the image of $A_t$ by the transition from $t$ to $t''$. Conversely, if $B_t$ transitions to a configuration $B_{t'}$ at $t'$ and to a configuration $B_{t''}$ at $t''$, then $B_{t'}$ must transition to a configuration at $t''$, and this configuration is $B_{t''}$. Similarly for the links.
\end{udefn}
\hypertarget{references}{}\subsection*{{references}}\label{references}
[[Blog - evolution and categories]]
T. F. H. Allen and T. B. Starr, \href{http://en.wikipedia.org/wiki/Special:BookSources/9780226014326#Online_text}{Hierarchy: Perspectives for Ecological Complexity}. Chicago: University of Chicago Press, 1982, p. 326. \hyperlink{bAllenStarr1982}{$\hookleftarrow$}
A. J. Arnold and K. Fristrup, \href{http://www.empseb.unibe.ch/pdf/Arnold1982_Rankin.pdf}{The theory of evolution by natural selection: a hierarchical expansion}, \href{http://www.jstor.org/stable/2400448}{Paleobiology}, vol. 8, no. 2, pp. 113–129, 1982. \hyperlink{bArnoldFristrup1982}{$\hookleftarrow$}
A. C. Ehresmann and J. P. Vanbremeersch, \href{http://en.wikipedia.org/wiki/Special:BookSources/9780444522443#Online_text}{Memory Evolutive Systems; Hierarchy, Emergence, Cognition}, Volume 4 (Studies in Multidisciplinarity). Elsevier Science, 2007, p. 402. \hyperlink{bEhresmann2007}{$\hookleftarrow$}
G. L. Farre, The Energetic Structure of Observation: A Philosophical Disquisition, \href{http://dx.doi.org/10.1177/0002764297040006004}{American Behavioral Scientist}, vol. 40, no. 6, pp. 717-728, May. 1997. \hyperlink{bFarre1997}{$\hookleftarrow$}
S. A. Frank, \href{http://stevefrank.org/abstracts/95JTB-Price.html}{George Price’s contributions to evolutionary genetics}., \href{http://dx.doi.org/10.1006/jtbi.1995.0148}{Journal of theoretical biology}, vol. 175, no. 3, pp. 373-88, Aug. 1995. \hyperlink{b1Frank1995}{$\hookleftarrow^1$} \hyperlink{b2Frank1995}{$\hookleftarrow^2$}
S. A. Frank, \href{http://stevefrank.org/foundations/foundations.pdf}{Foundations of social evolution}. \href{http://en.wikipedia.org/wiki/Special:BookSources/9780691059341#Online_text}{Princeton Univ Press}, 1998. \hyperlink{bFrank1998}{$\hookleftarrow$}
S. Okasha, \href{http://en.wikipedia.org/wiki/Special:BookSources/9780199267972#Online_text}{Evolution and the levels of selection}. New York: Oxford University Press, USA, 2006. \hyperlink{bOkasha2006}{$\hookleftarrow$}
G. R. Price, \href{http://www.cs.ucl.ac.uk/staff/ucacbbl/ftp/papers/price_nature.pdf}{Selection and Covariance}, \href{http://dx.doi.org/10.1038/227520a0}{Nature}, vol. 227, no. 5257, pp. 520-521, Aug. 1970. \hyperlink{bPrice}{$\hookleftarrow$}
G. R. Price, \href{http://www.dynamics.org/Altenberg/LIBRARY/REPRINTS/Price_extension_AnnHumGenetLond.1972.pdf}{Extension of covariance selection mathematics}, \href{http://dx.doi.org/10.1111/j.1469-1809.1957.tb01874.x}{Annals of Human Genetics}, vol. 35, no. 4, pp. 485-490, Apr. 1972. \hyperlink{bPrice}{$\hookleftarrow$}
G. R. Price, The nature of selection, \href{http://dx.doi.org/10.1006/jtbi.1995.0149}{Journal of Theoretical Biology}, vol. 175, no. 3, pp. 389-396, Aug. 1995. (written ca. 1971 and published posthumously) \hyperlink{bPrice}{$\hookleftarrow$}
H. A. Simon, \href{http://www.eecs.ucf.edu/~dcm/Teaching/COP4600-Spring2011/Literature/Complexity-Simon.pdf}{The architecture of complexity}, \href{http://www.jstor.org/stable/985254}{Proceedings of the American Philosophical Society}, vol. 106, no. 6, pp. 467–482, 1962. \hyperlink{b1Simon1962}{$\hookleftarrow^1$}\hyperlink{b2Simon1962}{$\hookleftarrow^2$}
H. A. Simon, \href{http://www.google.com/url?sa=t&source=web&cd=4&ved=0CC4QFjAD&url=http%3A%2F%2Foctopus.library.cmu.edu%2Fcgi-bin%2Ftiff2pdf%2Fsimon%2Fbox00021%2Ffld01496%2Fbdl0001%2Fdoc0001%2Fsimon.pdf&ei=8HA3TsryHKfy0gGKoL3bAw&usg=AFQjCNG7l94z8pVUPhR5-MkS1mFTDGLyyg}{Near decomposability and the speed of evolution}, \href{http://dx.doi.org/10.1093/icc/11.3.587}{Industrial and Corporate Change}, vol. 11, no. 3, pp. 587-599, Jun. 2002. \hyperlink{b1Simon2002}{$\hookleftarrow^1$}\hyperlink{b1Simon2002}{$\hookleftarrow^2$}\hyperlink{b3Simon2002}{$\hookleftarrow^3$}
J. Maynard Smith and E. Szathmáry, \href{http://en.wikipedia.org/wiki/Special:BookSources/019850294X#Online_text}{The major transitions in evolution}. New York: Oxford University Press, USA, 1995. \hyperlink{bSmithSzathmary1995}{$\hookleftarrow$}
[[Blog - evolution and categories]]
category:blog,biology
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