What is chemodiversity?

It was observed, that many plants seem to produce many compounds with no obvious purpose
Using resources to produce such compounds (instead of i.e. growing) should yield a fitness-disadvantage
one expects evolution to eliminate such behavior

Question: Why is this behavior observed?

Are these compounds necessary for some unresearched reason?
- unknown environmental effects?
- unknown intermediate products for necessary defenses?
- speculative diversity because they could be useful after genetic mutations?

Screening Hypothesis

First suggested by Jones & Firn (1991)
new (random) compounds are rarely biologically active
plants have a higher chance finding an active compound if they diversify
many (inactive) compounds are sustained for a while because they may be precursors to biologically active substances

There are indications for and against this hypothesis by various groups.

Setting up a simulation

If you wish to make apple pie from scratch, you must first create the universe
- Carl Sagan

Defining Chemistry

First of all we define the chemistry of our environment, so we know all possible interactions and can manipulate them at will.
We differentiate between Substrate and Products:
- Substrate can just be used (i.e. real substrates if the whole metabolism should be simulated, PPM^[1] in our simplified case)
- Products are nodes in our chemistry environment.

In Code:

data Compound = Substrate Nutrient
              | Produced Component
              | GenericCompound Int

Usage in the current Model

The Model used for evaluation just has one Substrate:
PPM with a fixed Amount to account for effects of sucking primary-metabolism-products out of the primary metabolic cycle
This is used to simulate i.e. worse growth, fertility and other things affecting the fitness of a plant.
We are not using named Compounds, but restrict to generic Compound 1, Compound 2 …
Not done, but worth exploring:
- Take a “real-world” snapshot of Nutrients and Compounds and recreate them
- See if the simulation follows the real world

Defining a Metabolism

We define Enzymes as
- having a recipe for a chemical reaction
- are reversible
- may have dependencies on catalysts to be present
- may have higher dominance over other enzymes with the same reaction
Input can be Substrate and/or Products
Outputs can only be Products
\(\Rightarrow\) This makes them to Edges in a graph combining the chemical compounds

Usage in the current Model

Enzymes all
- only map 1 input to 1 Output with a production rate of 1 per Enzyme
  (i.e. -1 Compound 2 -> +1 Compound 5)
- are equally dominant
- need no catalysts

Defining Predators

Predators consist of
- a list of Compounds that can kill them
- a fitness impact (\([0..1]\)) as the probability of killing the plant
- an expected number of attacks per generation
- a probability (\([0..1]\)) of appearing in a single generation
Predator need not necessary be biologically motivated
- i.e. rare, nearly devastating attacks (floods, droughts, …) with realistic probabilities

Example Environment

The complete environment now consists of
- Compounds:
- Enzymes:
- Predators:

Additional rules:

Every “subtree” from the marked PPM is treated as a separate species (fungi, animals, …)
\(\Rightarrow\) Every predator can only be affected by toxins in the same part of the tree
Trees can be automatically generated in a decent manner to search for environmens where specific effects may arise

Plants

A Plant consists of

a Genome, a simple list of genes
- Triple of (Enzyme, Quantity, Activation)
- without order or locality (i.e. interference of neighboring genes)
- Quantity is just an optimization (=Int) to group identical Activations
- Activation is a float \(\in [0..1]\) to regulate the activity of the Enzyme genetically
an absorbNutrients-Function to simulate various effects when absorbing nutrients out of the environment, depending on the environment (i.e. can use informations about chemistry, predators, etc.)
- Not used in our simulation, as we only have PPM as “nutrient” and we take everything given to us.

Metabolism simulation

Creation of compounds from the given resources is an iterative process:

First of all we create a conversion Matrix \(\Delta_c\) with corresponding startvector \(s_0\).
We now iterate \(s_i = (\mathbb{1} + \Delta_c) \cdot s_{i-1}\) for a fixed number of times (currently: \(100\)) to simulate the metabolism^[2].

^[2]: Thats a ‘lie’, we calculate \((\mathbb{1} + \Delta_c)^{100}\) efficiently via lapack-internals
Entries in the matrix come from the Genome: an Enzyme which converts \(i\) to \(j\) with quantity \(q\) and activity \(a\) yield \[\begin{eqnarray*} \Delta_c[i,j] &\mathrel{+}=& q\cdot a,\\ \Delta_c[j,i] &\mathrel{+}=& q\cdot a, \\ \Delta_c[i,i] &\mathrel{-}=& q\cdot a, \\ \Delta_c[j,j] &\mathrel{-}=& q\cdot a \end{eqnarray*}.\]
- This makes the Enzyme-reaction invertible as both ways get treated equally.

Metabolism-example

Given a simple Metabolism with \(1\) nutrient (first row/column) and \(2\) Enzymes in sequence, we have given \(\Delta_c\) wtih corresponding startvector \(s_0\): \[\Delta_c = 0.01 \cdot \begin{pmatrix} -1 & 1 & 0 \\ 1 & -2 & 1 \\ 0 & 1 & -1 \\ \end{pmatrix}, s_0 = \begin{pmatrix}\text{PPM:} & 3 \\ \text{Compound1:} & 0 \\ \text{Compound2:} & 0\end{pmatrix}.\]
In the simulation this yields us \[s_{100} \approx \begin{pmatrix}\text{PPM:} & 1 \\ \text{Compound1:} & 1 \\ \text{Compound2:} & 1\end{pmatrix},\] which is the expected outcome for an equilibrium.

Assumptions for metabolism simulation

All Enzymes are there from the beginning
All Enzyme-reactions are reversible without loss
static conversion-matrix for fast calculations (unsuited, if i.e. enzymes depend on catalysts)
One genetic enzyme corresponds to (infinitely) many real (proportional weaker) enzymes in the plant, which get controlled via the “activation” parameter

Fitness

We handle fitness as \(\text{survival-probability} \in [0..1]\) and model each detrimental effect as probability which get multiplied together.
To calculate the fitness of an individual we take three distinct effects into consideration:
- Static costs of enzymes
  - Creating enzymes weakens the primary cycle and thus possibly beneficial traits (growth, attraction of beneficial organisms, …) \[F_s := \text{static_cost_factor} \cdot \sum_i q_i \cdot a_i \quad | \quad (e_i,q_i, a_i) \in \text{Genome}\]
  - limits the amount of dormant enzymes
- Cost of active enzymes
  - Cost of using up nutrients \[F_e := \text{active_cost_factor} \cdot \frac{\text{Nutrients used}}{\text{Nutrients available}}\]
- Deterrence of attackers \(F_d\) (next slide)

Attacker

Predators are modeled after Svennungsen et al. (2007)
Each predator has an expected number of attacks \(P_a\), that are poisson-distributed with impact \(P_i\).
Plants can defend themselves via
- toxins that the predator is affected by with impact-probability \(D_t(P_i)\)
- herd-immunity via effects like automimicry: \(D_{pop} = \mathbb{E}[D_t(P_i)]\)
All this yields the formula:

\[F_d := 1 - e^{- (D_{pop} \cdot P_a) (1-D_t(P_i))}\]
The attacker-model is only valid for many reasonable assumptions
- equilibrium population dynamics
- equal dense population
- which individual to attack is independently chosen
- etc. (Details in the paper linked above)

Haploid mating

We hold the population-size fixed at \(100\)
Each plant has a reproduction-probability of \[p(\textrm{reproduction}) = \frac{\textrm{plant-fitness}}{\textrm{total fitness in population}}\] yielding a fitness-weighted distribution from that \(100\) new offspring are drawn
in inheritance each gene of the parent goes through different steps (with given default-values)^[3]

^[3]: in case of quantity \(q > 1\) the process is repeated \(q\) times independently.
- mutation: with \(p_{mut} = 0.01\) another random enzyme is produced, but activation kept
- duplication: with \(p_{dup} = 0.05\) the gene gets duplicated (quantity \(+1\))
- deletion: with \(p_{del} = p_{dup}\) the gene get deleted (or quantity \(-1\))
- addition: with \(p_{add} = 0.005\) an additional gene producing a random enzyme with activation \(0.5\) gets added as mutation from genes we do not track (i.e. primary cycle)
- activation-noise: activation is changed by \(c_{noise} = \pm 0.01\) drawn from a uniform distribution, clamped to \([0..1]\)

Simulations

Overall question: What parameters are necessary for chemodiversity?
- How can we see chemodiversity?
- We define an Enzyme \(E\) as divers, if the average of this Enzyme in the population stays below \(0.5\), so \(E_i \in E_{div} \text{iff.} \mathbb{E}[E_i] < 0.5\)
- We can then count the number of diverse Enzymes per plant \(E_{d,p_i} = |\left\lbrace E_i | E_i \in E_{div}, E_{i,p_i} > 0.5, \right\rbrace|\)
To get an insight into how this behaves we observe several other parameters every generation:
- Fitness \(\in [0..1]\)
- Number of different compounds created
- Amount of compounds created
- Number of Plants theoretically resistant to predator \(i\) (i.e. can produce a toxin to defend themselves, albeit not to \(100\%\).

Simulations (cont.)

General setup of the simulation:
- All using the example-environment shown before
  - 27 different compounds, 1 Nutrient (simulating the primary metabolism)
  - 7 of 27 compounds are toxic
  - at least 3 compounds are needed for total immunity
  - 4 predators
- Duration of 2000 generations
Different setups tested:
- Behavior of predators (AlwaysAttack, AttackRandom, AttackInterval Int)
- varying \(\text{static_enzyme_cost}\) from \(0.0\) to \(0.20\) in steps of \(0.02\)
  - effectively limits the amount of maximal enzymes to \(\frac{1}{\text{static_enzyme_cost}}\)
- varying \(\text{nutrient_impact}\) from \(0.0\) to \(1.0\) in steps of \(0.1\)
  - makes toxins less/more costly to produce

Results

It doesn’t matter how beautiful your theory is, it doesn’t matter how smart you are. If it doesn’t agree with experiment, it’s wrong.
- Richard P. Feynman