407 lines
14 KiB
Markdown
407 lines
14 KiB
Markdown
---
|
|
title: Chemodiversity
|
|
subtitle: A short overview of this project
|
|
author: Stefan Dresselhaus
|
|
license: BSD
|
|
affiliation: Theoretic Biology Group<br>
|
|
Bielefeld University
|
|
abstract: Attempt to find indications for chemodiversity in the plant secondary metabolism according to the screening hypothesis
|
|
date: \today
|
|
|
|
papersize: a4
|
|
fontsize: 10pt
|
|
documentclass: scrartcl
|
|
|
|
margin: 0.2
|
|
slideNumber: true
|
|
...
|
|
|
|
|
|
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](https://doi.org/10.1098/rstb.1991.0077))
|
|
- 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](https://nph.onlinelibrary.wiley.com/doi/full/10.1111/nph.12526#nph12526-bib-0093).
|
|
|
|
|
|
--------------------------------------------------------------------------------
|
|
|
|
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`{.haskell}** and
|
|
**`Products`{.haskell}**:
|
|
- **`Substrate`{.haskell
|
|
}** can just be used (i.e. real substrates if the whole metabolism
|
|
should be simulated, **`PPM`{.haskell}**^[1]^ in our simplified case)
|
|
- **`Products`{.haskell
|
|
}** are nodes in our chemistry environment.
|
|
- In Code:
|
|
```haskell
|
|
data Compound = Substrate Nutrient
|
|
| Produced Component
|
|
| GenericCompound Int
|
|
```
|
|
::: footer
|
|
^[1]^: plants primary metabolism
|
|
:::
|
|
|
|
|
|
Usage in the current Model
|
|
--------------------------
|
|
|
|
- The Model used for evaluation just has one `Substrate`{.haskell}:
|
|
`PPM`{.haskell} 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`{.haskell}, `Compound 2`{.haskell} ...
|
|
- 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 **`Enzyme`{.haskell}s** 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`{.haskell} and/or `Products`{.haskell}
|
|
- Outputs can only be `Products`{.haskell}
|
|
- $\Rightarrow$ This makes them to Edges in a graph combining the chemical
|
|
compounds
|
|
|
|
Usage in the current Model
|
|
--------------------------
|
|
|
|
- `Enzyme`{.haskell}s all
|
|
- only map `1`{.haskell} input to `1`{.haskell} Output with a production rate of `1`{.haskell} per `Enzyme`{.haskell}
|
|
(i.e. `-1 Compound 2 -> +1 Compound 5`{.haskell})
|
|
- are equally dominant
|
|
- need no catalysts
|
|
|
|
Defining Predators
|
|
------------------
|
|
|
|
- **`Predator`{.haskell}s** consist of
|
|
- a list of `Compound`{.haskell}s 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`{.haskell} need not necessary be biologically motivated
|
|
- i.e. rare, nearly devastating attacks (floods, droughts, ...) with realistic
|
|
probabilities
|
|
|
|
Example Environment
|
|
-------------------
|
|
|
|
:::::::::::::: {.columns}
|
|
|
|
::: {.column width=37%}
|
|
|
|
- The complete environment now consists of
|
|
- `Compound`{.haskell}s:
|
|
{style="vertical-align:middle"}
|
|
- `Enzyme`{.haskell}s:
|
|
{style="vertical-align:middle"}
|
|
- `Predator`{.haskell}s:
|
|
{style="vertical-align:middle"}
|
|
|
|
:::
|
|
|
|
::: {.column width=63% .fragment}
|
|
|
|
{width=75%}
|
|
|
|
Additional rules:
|
|
|
|
- Every "subtree" from the marked `PPM`{.haskell} 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
|
|
:::
|
|
|
|
::::::::::::::
|
|
|
|
::::: notes :::::
|
|
|
|
CTRL+Click for zoom!
|
|
|
|
- All starts at PPM (Plant Primary Metabolism)
|
|
- Red = Toxic
|
|
- Blue = Predators
|
|
|
|
::::
|
|
|
|
--------------------------------------------------------------------------------
|
|
|
|
Plants
|
|
------
|
|
|
|
A **`Plant`{.haskell}** consists of
|
|
|
|
- a **`Genome`{.haskell}**, a simple list of genes
|
|
- Triple of `(Enzyme, Quantity, Activation)`{.haskell}
|
|
- without order or locality (i.e. interference of neighboring genes)
|
|
- `Quantity`{.haskell} is just an optimization (=Int) to group identical
|
|
`Activation`{.haskell}s
|
|
- `Activation`{.haskell} is a float $\in [0..1]$ to regulate the activity of
|
|
the `Enzyme`{.haskell} genetically
|
|
- an `absorbNutrients`{.haskell}-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`{.haskell} 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]^.
|
|
|
|
::: footer :::
|
|
^[2]^: Thats a 'lie', we calculate $(\mathbb{1} + \Delta_c)^{100}$ efficiently via
|
|
`lapack`-internals
|
|
:::
|
|
|
|
- Entries in the matrix come from the `Genome`{.haskell}: an `Enzyme`{.haskell} 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)](http://doi.org/10.1098/rspb.2007.0456)
|
|
- 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]^
|
|
|
|
::::: footer
|
|
^[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 $[0..1]$
|
|
|
|
:::: notes
|
|
- Default values **not** motivated in any way!
|
|
- finding out how these values influence is core!
|
|
::::
|
|
|
|
--------------------------------------------------------------------------------
|
|
|
|
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`{.haskell}, `AttackRandom`{.haskell}, `AttackInterval Int`{.haskell})
|
|
- 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
|
|
|
|
--------------------------------------------------------------------------------
|
|
|
|
Effect of Predator-Behavior onto chemodiversity
|
|
----------------------------------------
|
|
|
|
|
|
|
|
Effect of static enzyme cost
|
|
----------------------------
|
|
|
|
|
|
|
|
Effect of static enzyme cost (cont.)
|
|
------------------------------------
|
|
|
|
|
|
|
|
Effect of static enzyme cost (cont.)
|
|
------------------------------------
|
|
|
|
|
|
|
|
Effect of nutrient-impact
|
|
-------------------------
|
|
|
|
|
|
|
|
Effect of nutrient-impact (cont.)
|
|
---------------------------------
|
|
|
|
|
|
|
|
Effect of nutrient-impact (cont.)
|
|
---------------------------------
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|