June 23, 1998

Modelling Algal Metabolism

outline

1. Description of the basic processes

2. Dynamic model

• simple kinetics at each step - tends to oscillate
• pools, threshholds and sigmoidal kinetics (modelling tricks)

3. Underlying system

• NK model

4. What does the oscillation tell us? Is it real

1. Generalized model of algal metabolism

This paper will focus on a very simple version of algal response to changes in light. The underlying conceptual model will be derived Shuter's work (1975?) on optimal physiological composition. His model has 5 major components (see table 1) that exist in different relative amounts depending on the algal growth limitations. According to Shuter, the optimal solution is the combination of components that gives the maximum growth rate. Part of the optimization of these systems is a series of feedback loops that maintains the organism in "balanced growth", i.e. where the specific rate of increase of all components is equal. I will explore this model with a dynamic model constucted to illustrate this feedback control and with a complex-system model that is constructed to understand the basic characteristics of the feedback system.

Table 1. Cellular components as defined by Shuter (1975)

From the literature and experience we can describe a "typical yet non-existent" algal metabolism that is based on these components. Energy is captured by the photosynthetic membranes to make NADPH. NADPH is used by the photosynthetic enzymes to fix CO2 into Triose. Triose is converted into building blocks for synthesis and Triose is also used for respiration processes to support biosynthesis, growth and maintainence costs. The building blocks are assembled into the four cellular components. The allocation between these different components represents the fundamental adaptation strategy for the organism.

Figure 1. Diagram of carbon flow through algal metabolism. The components are described in Table 1. The feedback control loops are described below. NADPH represents the energy output of the photosythetic membrane. Triose-P represents the carbon fixed into organic carbon. Blocks represent the common building blocks used for biosynthesis including carbohydrates, amino acids, nucleotices and fatty acids.

I can tell a story that describes how this system is controlled that is consistent with our understanding of the regulation of photosynthesis but doesn't include an details at the molecular level.. The rate of production of NADPH depends on light energy and the amount of photosynthetic membrane (Pmemb). High amounts of NADPH itself should inhibit the production. Triose-P is produced depending on the amount of photosynthetic enzyme (Penz) and the amount of NADPH to fuel the reaction. Triose production should be inhibited if there is already a large concentration of building blocks. Building blocks are produced from carbon skeletons and nitrogen. These common metabolites are synthesized by enzymes and as a result the rate is a function of the enzymatic and biosynthetic component. These building blocks are then synthesized into the biomolecules of the cell. Amino acids are synthesized into proteins at a rate controlled by the ribosomes and thus the rate of all of these processes are sensitive to the Enz fraction. The concerted regulation of cell metabolism and biosynthesis should lead to a set of components (Pmemb, Penz, Enz and Struct) that no component is in excess, i.e. where a component is synthesized in excess of its amount necessary to mediate the metabolic process. The structural component is synthesized in a fixed ratio to cell growth and is not catalytically active. In Shuter's version of this story, each component is only synthesized in an amount that just meets the catalytic need and that will lead to an "optimal" growth rate. In the dynamic model of this process, the regulation of the flux of new carbon into each component will respond, through feedback loops, to converge on the optimal ratio of these components for a fixed environmental conditions and resources.

Steady state environmental conditions are rare. How do algae adapt to fluctuations in natural systems. There are several possible strategies; storage, rapid adaptation, cell-to-cell diversity, and metabolic diversity. These will be dealt with in another paper ("Do phytoplankton change their strategy in a fluctuating environment?"). This paper will deal with the regulatory system and the behavior of simple simulated models of this system. I hope to demonstrate that dynamic models can be constructed the reach the same opitmal solution as predicted by Shuter but that these models have some built in propensity to oscillate. We can damp that oscillation in the model with tricks that may help us understand how this works in a cell. If we characteize the regulatory circuit in its crudest logical form, using Boolean operators and simple time steps, we also observe oscillatory behavior that can be characterized as periodic attractors in two different basins. Comparison of the dynamic simulation model with the complex-system model should provide some insight into the underlying regulatory strategy of algae.

more here later.

2. Dynamic simulation model

2a. Constructing a simple dynamic model using STELLA

Figure 3. Model schematic in STELLA. Stocks are the boxes, flows are the arrows with a circle in the middle and information flow are the single lines.

Table 2

figure 3 output from a simple model

2b. Adding pools, threshholds and sigmoidal kinetics as modelling tricks

Table 3. "Improved" equations for a well-behaved model

Figure 4. Output from the "improved" model

Comments on these tricks as they relate to algal regulation.

3. Complex model of this same metabolic system

3a. Boolean statements

In each case the value for the metabolite of component is determined by the state of the system in the previous time step.

3b. Logic table for output

There are 128 (2^7) different combinations of 1 and 0 for this system. I have constructed a logic table in Excel that determines the outcome for each initial state. The outcome can be given in binary (i.e. 100100) or decimal (36). For each initial state their is only one outcome state and thus we can map 128 different inputs onto each of their outputs. Following the progress of these states will illustrate the concept of a basin. If we start at state 0 it goes to 1, 1 goes to 15, 15 to 79 to 111 to 127 to 110 to 113 to 14 to 65 and back to 15. There is a loop in here that is referred to as a periodic attractor for this system. It turns out that in this system there are two periodic attractors and that all of the possible inputs are in a basin for one or the other attractor. A slight change in the state of one component can result in the system going into the other basin. For example, starting at 23 --> 14 which is in Basin A but starting at 24 --> 70 -->33 --> 31 which is in Basin B. This is the essense of non-linear systems, a slight change does not necessarily lead to an incremental change in the output.

It is easier to visualize the concept of states flowing into a basin from Figure 5.

Figure 5.Basin and attractor diagram for experimental run "try3". Basin A and Basin B are the periodic attractors. The 128 states are listed as their decimal equivalent for the binary values from 0000000 to 1111111. This is the same data as in Appendix A.

3d. Effect of a slight change in the logic statements

Appendix A.

Logic table for "try3"