Draft for Algal Physiology Course

February 02, 2002

Examining the regulation of algal metabolism:
three perspectives

John Rueter

Key words:

modeling, regulation, dynamic models, descriptive models, complex models

Abstract:

:The patterns and processes of algal growth have been studied with observation and modeling. Through many observations of different species and conditions we appreciate general patterns of response. Descriptions of these can include mechanisms and regulatory logic that control the response at any one time. Dynamic models can be used to simulate the sequence of regulation and growth that would result from these controls. Dynamic models are very useful in exploring the interactions and feedback that might occur. Dynamic models are often numerical, time step simulations, and are built to represent the same underlying regulatory logic as the descriptive, verbal models. From any description of algal response there are many valid choices for the logic to use in these dynamic models and thus the generality of any one set of logics is in question. This paper introduces another set of models that can be used to look at the regulatory logic explicitly to examine the patterns that would be expected just from the structure of the regulatory logic, and independent of the parameratization of the dynamic models. This class of models is based on the NK Boolean networks as described by (Kaufmann 1994) and allows for the comparison of multiple logical constructions. The purpose of this paper is to attempt to validate the logical choices made for dynamic models and to look for how the three types of models inform each other.

Table of contents:

1. Introduction

2. Conceptual model: The verbal "story" of algal metabolism

3. Numerical model for metabolism and adaptation

4. Modeling the metabolism as a complex network

5. Comparison of the three models

5.1 What the STELLA model tells us about the Boolean NK model

5.2 What the Boolean NK model tells us about the STELLA model

5.3 Limitations and strengths of the verbal model

6. Conclusions

7. References

8. Links to appendices

1. Introduction

Trying to describe how algae work and respond to environmental parameters is a important venture. Algae are crucial in food chains, geochemical cycles and the evolution of life on earth. There have been good studies on the optimization of algal metabolism that describe how the composition of algae would be changed to make optimal use of resources (Shuter 197*, and others) or comparing broad groups of algae for their responses to the environment (Reynolds CSR framework). These studies help us understand the limits to productivity and how different cellular composition or structure helps species compete for resources. This paper attempts to help us understand how cells get to the optimal composition, as described by Shuter (197*), i.e. how they regulate their minute-to-minute metabolism in order to change their composition.

In our attempt to build models of algal metabolism and regulation we have three parts to combine (Flake 2001): the environment (in this case the external parameters and the algal biochemistry and physiology), the model of the environment (our simplified description of the objects and how they respond) and our search strategy for finding a model that describes the environment. In this effort, our search strategy resulted in three different models. None of these models are any "better" than the other, and each has its advantages for understanding the processes.

This paper attempts to describe the physiological response to light with three different approaches, each based on a different underlying model. These are totally different constructions not meta-models i.e. they are not different algebraic forms of the same general model (see Burmaster ****) The three approaches are:

1. The descriptive approach can be rich in detail and link to photographs, diagrams or other visualizations of the organization and function of algae. Different species can be easily compared for their responses or structure. A good example of this type of model is the PEG*- community succession model of Sommer et al. (1986).
2. The systems view of algal metabolism is very structured. The version that we will use will be based on the flow of energy/carbon through the cell pathways. An important feature the emerges from this approach are the common regulatory motifs, such as flow limitation and feedback inhibition. A good example of this type of model is the "dynamic balance hypothesis" model of Kana et al. (1997).
3. The complex/network approach examines the metabolism as a connected network of nodes that all effect each other directly or indirectly. A behavior for the system emerges that has fundamental characteristics related to the nature of the network and the connectedness of that network. This view emphasizes the network aspect of metabolism. There are no examples of this type of model applied to algae that I know of .

I will address each of these approaches separately and link both the systems view and the network view back to the underlying conceptual model. That will be fairly straightforward since each of these models were originally derived to fit the conceptual model. In each case I will explain how the systems view or the network view increases our understanding of how algae work. I will also attempt to compare the systems view to the network view. This will be less straightforward and more theoretical but, I hope, will help validate the use of these other modeling approaches.

2. Conceptual model: The verbal "story" of algal metabolism and regulation

There is a generally accepted story for the response of algae to light. The details change with the species. The magnitude of the responses and plasticity of the metabolism is different, but it is essentially the same pattern. I will explore one version of this pattern and compare it to a version with one slight modification.

The basic assumption is that algae respond to light (and other environmental factors) through detecting changes in the balance of their internal metabolism that are caused by changes in light. This means that the algae don't respond directly to light with photoreceptors that control metabolism, but that their response is cued by relative increases or decreases in the light generated metabolic intermediates (Kana et al ****). The actual regulation is certainly more complicated than this, but this simplification allows us to proceed.

In response to either changes in the external environment or changes in the internal balance of metabolic intermediates and metabolic rates, the cell will produce more or less intermediate metabolic intermediates and more or less components of the cell. This is the two-component model, that accounts for control of both metabolism (at the time scale of minutes) and cell composition through protein synthesis (at the time scale of hours) as described Tandeau de Marsaac et al (*****). The metabolic intermediates (and abbreviations) we will address are:

NADPH as the direct output of the light reactions,

Triose-P as a proxy for the output of the dark - enzymatic reactions of photosynthesis,

"building blocks" as a descriptive term for the total assemblage of carbohydrates, amino acids, fatty acids and nucleotides.

The cell can also be divided into four components that are responsible for different aspects of cell physiology. These are essentially the same as described by Shuter (197*):

Pmemb -the photosynthetic membranes,

Penz - the photosynthetic enzymes including RUBISCO and the other RPPP enzymes,

Enz- all of the biosynthetic machinery including enzymes, respiration machinery, mitochondria and ribosomes

Struct - all the structural components of the cell including DNA and the cell wall.

This verbal model describes how the flow of energy and carbon skeletons are regulated as they move through the photosynthetic system and the biosynthetic machinery and how that carbon is reinvested in those cellular systems themselves. This flow could be represented by a typical "blackboard" sketch that uses arrows for flow and oversimplified representations of the cellular components.

The verbal model for the optimization of algal metabolism that we will base our story has been described by Shuter (197?). In the description on algal metabolism and composition we will use the terms "high" or "low" amounts to represent relative amounts of metabolic intermediates or functional components. Our story about regulation can be told in seven lines:

1. NADPH will increase if there is "high" Photosynthetic membrane that leads to its production, or if Triose-P is high enough to cause feedback inhibition of the use of NADPH. Thus one factor that may cause NADPH to increase in the next time period is if Pmemb had just increased in the last time period.
2. Triose-P will increase if there is a high amount of Penz and at the same time a high amount of NADPH to fuel the reactions for the Calvin Cycle. Triose-P may also be high because the amount of Blocks were so high that they caused feedback inhibition of the use of Triose-P.
3. Building blocks will increase if there was simultaneously high Triose-P and high Enzymatic and biosynthetic machinery to provide the respiratory power and catalysis for biosynthesis.
4. Photosynthetic membranes will increase if the amount of NADPH is low and if there is a high amount of Blocks to support new synthesis.
5. Photosynthetic enzymes will increase if there is a low amount of Triose-P and sufficient building blocks to support synthesis.
6. The enzymatic and biosynthetic machinery in the cell will increase if the building blocks are high.
7. The structure will increase if the building block concentration is high.

This verbal model for cell regulation should lead to an composition of the cell that will produce an optimal amount of new biomass per time. The convergence on this optimum will happen because if there is too much or too little of a component (Pmemb, Penz, Enz, or Struct) the cell stops making that component and puts new material into the component that has the least relative output. For example, if there is too much Pmemb compared to Penz, the NADPH will be high and the Triose will be low. The cell will respond by making more Penz. The other components are regulated in a similar manner.

Remember this story represents one of the myriad of logical connections that could be seen in different algal species. Later we will discuss a slightly changed regulation logic that provides another alternative for the control of Penz. This change allows for the additional possibility to increase Penz if high NADPH concentrations build up, because it isn't being used up fast enough.

(line 5) can be changed from

Penz will increase if there is a low amount of Triose-P and high building blocks

to

Penz will increase if there is either low Triose-P and building blocks, or there is high NADPH and building blocks

Our verbal model is very useful in describing relationships between the metabolic intermediates and components and to get a "feel" for the generalized response of algae. It is difficult to examine the interactions between different components that may not be directly linked by a statement in our seven lines. For example, there is no direct statement that ties the production of Pmemb to that of Penz but you can tell from our previous example that some conditions would cause Pmemb to increase while Penz was held constant.

3. Numerical model for metabolism and adaptation

Algal metabolism can be considered as a flow of carbon from photosynthesis through central metabolism to increases in the cellular components that make up the algal biomass. This type of flow and flow regulation is amenable to modeling in the simulation language of STELLA. In STELLA the concentration of metabolic intermediates and components are represented by boxes, the flow of material is represented by the open arrows with the circle in the middle, and information flow is represented by the single line arrows. These icons for particular functions in systems modeling helps structure the model and make us be more explicit about the relationships.

Figure 2: STELLA model for algal metabolism. The metabolic intermediates are represented by the three boxes on the left. These are fed and connected by flows that depend on the amount of the biochemical components (the column of boxes on the right) and environmental input. Information flow is determined by the thin red lines. Cell growth is represented by the reinvestment of cellular building blocks into the cellular components. The regulatory logic is determined by the structure of the information flows in this model and the equations that are hidden. See appendix 1 for the full list of equations and appendix 2 for example outputs.

Each metabolite and component is assigned an initial value. The flows between these boxes are controlled by input from metabolite and component boxes. In this simple model, the flows are controlled by threshold type relationships. For example, the flow from NADPH to Triose-P is controlled by the equation

This equation sets a threshold (1) for the product that results in total feedback inhibition and otherwise allows the forward reaction to proceed as a simple saturating process that depends on the amount of Penzyme catalyst and the amount of NADPH as a substrate. The other reactions in this model are constructed similarly, please see Appendix 1 for the full program listing.

The value of examining this set of relationships in STELLA is that it takes the current value of each component and uses it to calculates the outcome for the next time period. STELLA also provides useful visualization tools such as graphs and tables. In addition, while you are running the model, it updates the graphs each time period resulting in an animation of the relationships that can help visualize and understand the dynamics of the system. Even with only seven components and very simple mathematical relationships the behavior of this system is very dynamic, exhibiting wide swings in concentrations and oscillations in the metabolite and component concentrations. (These dynamics can be damped by using modeling tricks that may or may not relate to algal physiology as I will explore later.)

One aspect of algal metabolism that can be demonstrated from this model is the nature of adaptation through plasticity of composition. The initial conditions have been picked arbitrarily. If the alga is exposed to different light input values, the cell responds on a timestep-to-timestep that regulates the metabolism, but this results in a reallocation strategy that is apparent over 100 time steps.

 

 

Figure 3. Response of the STELLA model to light. a) light level of 10 results in higher Pmemb and lower Enz. b) a light level of 200 results in lower Pmemb and more Enz.

redo these figures

Another important aspect of algal metabolism that can be demonstrated from such a dynamic model is the value of adaptation. The variation in the composition results in better response to the available resource in the sense that the cell is better able to grow on that level of resource after it has changed its composition. Another way to look at this is that as the resource becomes more scarce the cells are able to grow relatively better at lower levels, for example a decrease in light by 50% that would result in a 50% decrease in photosynthesis in an unadapted cell results in only a 13% decrease in an adapted cell (Figure 4).

Thus the dynamic model demonstrates and helps visualize the interactions between the components and metabolic intermediates in both the timestep-to-timestep flow and the accumulation of differences that lead to adaptation over longer time. Comparing the effect of a slight change in the regulation is possible both in the gross response to light and in the minute-to-minute pattern of regulation. The alternate regulation proposed for the conceptual model can be included in this STELLA model by the addition of another arrow and rewriting the flow between Blocks and Penz.

Figure 5. Change in logic for the STELLA models. The new logic requires the concentration of NADPH as an addition input to the contol over Penz component.

Letting each case of the model adapt for 100 hours to a range of light conditions results in different biomass vs. light curves Figure 6. These differences can be explained in terms of the logic modification made, the second version has two options for increasing the Penz and shows a wider divergence in the Pmemb to Penz ratio. This allows the cells to have more to invest into the Enz fraction, which in turn allows faster "growth". The two versions of the model have the same photosynthesis up to L=75 and then at higher light, the second version has a higher productivity.

Figure 6. Comparison of P vs. I curves for the first logic set (try5) and the second logic set (try6). Each model was allowed to adapt to each light intensity for 100 hours.

Even with the limited number of metabolic intermediates and components in this STELLA model, there are many other modifications that could be made that would still be consistent with our original verbal model. It is possible that other versions of this model, even with the same general flow pattern, could have very different behavior. In some cases it can be shown that these different instances of the information flow result in the same optimal component composition under steady state conditions, but the path and time to achieve optimal composition are different. Thus optimization models, such as those described by Shuter (197*), are necessary but not sufficient to describe competition in variable environments. As shown in Figures 4 and 6, the regulation logic can have major impact on the net productivity after a change in conditions. Given that the regulation logic in cells depends on the wiring (connections made for information flow) rather than the efficiency of the appliances (the components),

The current paradigm for studying adaptation and competition is built on the "modern synthesis", essentially a telelogical argument that says that efficient proteins lead to competitive advantage, competitive advantage leads to higher proportion of efficient genes in the population, and these genes lead back to the efficient proteins. This paper presents an additional way to look at competition in a variable environment that would depend more on the logic of the regulation than the efficiency of the proteins. Changes in regulatory structure could happen with very little change in genetic code. If the connections for information flow are so important then we are faced with the task of understanding the underlying logic of the system independent of initial conditions, threshold levels, rates of flow and response patterns. In a sense, we are trying to separate out the grammar of the control structure from the numerical relationships. In the next section, I describe a method for looking at the logic of the control system.

4. Modeling the metabolism as a complex network

Boolean NK to study the underlying logic of the grammar of regulation (Kauffman 1994)

** phrases from Flake **

Flake (2001)

A third way to look at the metabolism of algae is as a set of seven nodes that are connected with simple logical relationships. I have translated the conceptual description (lines 1-7) into Boolean relationships and present them here. In each case the value of the node in the second time step depends on the values of one or more nodes in the first time step. This approach will allow me to analyze the nature of the network, without variations in thresholds and response functions. In particular, this approach is well suited to looking for causes of oscillations similar to what we saw above in the STELLA model output.

Boolean NK network for the algae metabolism model

There are 128 different combinations of true and false (or 1 and 0) for these 7 nodes and thus it is possible to exhaustively test all cases. This can be done in EXCEL by using Boolean logic operators. The results might look like this table

Table 1. Example Boolean state for the alga.

Which would mean that NADPH, Blocks, Penz and Enz increased or were "high" in the current time step and that Triose, Pmemb and Struct were "low" or didn't increase. Given these conditions we can calculate what would happen in the next time step.

Table 2. Status of the cell in the next time step and physiological explanation. See the list of Boolean equations above for the logical reason. Starting from the state given in Table 1.

We can keep track of the 128 possible starting points and 128 possible outcomes. We can also use a decimal equivalent for for each of those inputs as a type of short hand, for example 1111111(base 2) equals 127 (base10). For each input there is an output. The interesting part of this is to follow the chain of inputs that lead to outputs, make that the new input and proceed until a pattern emerges. For example, in this model state 76 leads to 96 leads to 64 leads to 0 and 0 leads to 0 again. This means if you start at 76 you will end up at 0 and stay at 0. In complex systems this is called an attractor. It turns out that this NK network has only three possible outcomes (attractors) 0, 115 and a cycle between 80 to 39 and back to 80 (called a periodic attractor). All of the values that lead to a particular outcome are said to be in one basin.

Table 3. Basins for the first model.

.

These basins are often visualized by drawing connections between all the numbers that eventually lead to a particular outcome. This shows more structure than is given above because not all 48 initial states lead directly to 0 but rather lead to other states that eventually lead to 0.

.

Figure 7. The original Boolean-NK model as described above results in three basins, two point attractors (0 and 115) and one periodic attractor (80<-->39). The average distance from any starting condition to the final attractor is 1.98 transitions. After the initial condition, 66% of the states are expressed. These parameters indicate the regulation doesn't immediately converge on the final outcome and that there are many different paths that lead to these final attractor states.

These values mean something:

If we make the same variation on the model as we did for the conceptual and the STELLA version, we can compare the differences in the output. Line 5 in our logic table should be changed to

5'. (NOT(Triose) AND Blocks) OR (NADPH AND Blocks)

If we analyze the same set of input and output tables as for the initial model we see slight differences in the behavior of the model, including one slight shift in a basin and a new periodic attractor as part of the third basin.

0 same as above

119 NADPH, Triose, Blocks, Enz and Blocks and Penz are high but Pmemb is low. Again this is a direct consequence of the change in the logic that provided another path for Penz to increase and is a possible metabolic outcome where Pmemb is providing a smooth flow of NADPH to the Calvin Cycle without high amounts of pigments and membrane material.

39-89 basin. same as above

103 NADPH, Triose, Penz, Enz and Struct are high but Blocks and Pmemb are low

112 NADPH, Triose and Blocks are high and all of the other components are low

This could be a reasonable oscillation that represents the central role of Blocks in regulating this network. If blocks are high, there is synthesis of Penz, Enz and Structure and if blocks are low there is no synthesis of these.

The above approach to examining the regulation model demonstrates that the basic logical structure of the network leads to a set of attractors and that if we change the logical network slightly we get a slightly different set of attractors. The interesting point here is that the behavior changes and that when you look at the number of attractors it seems like an incremental or gradual change, but if you pick a particular initial state the different outcome can be dramatic. This is the nature of complex or nonlinear systems: a small change in the input does not necessarily lead to a small, incremental change in the output. For example, in these two versions of the model there are 28 initial cases that will feed into the a new basin with the revised logic that otherwise would have been in the 39-80 basin in the original model logic.

5. Comparison of the three models

5.1 What the NK model tells us about STELLA

Many of the runs of the STELLA models show wide oscillation in the concentration of metabolic intermediates and components. Is this type of oscillation to be expected based on the underlying logical structure of the regulatory network? To address this question, we can examine some particular examples of output from both the original and the revised model.

One of the most dramatic examples is the revised model running at light = 100 (Figure c)/ After an original period of oscillation until about 30 hours there is clear linear responses as the building blocks increase. Then at about 40 hours these concentrations continue with the same trend but oscillate. Then at a little after 50 hours there is a pattern of broadening oscillation.

If we examine the output using a similar approach as for the Boolean NK network we can see that the states are changing in a manner that is similar (but not exactly like) the Boolean NK model. From 0 to 10 hours the response is hits only 4 different states. Up to 30 hours only a few more states are hit (out of the possible 128). At 30 hours the pattern makes an abrupt change to state 15 and then a mixture of 15 and 31 exclusively until 40 hours. From 40 to 55 hours it cycles between several values but hits 47 and 91 for a large portion of the time. After 55 hours the model goes into a region rich in 71 and 87 in varying ratios until 100 hours.

The values that this model hits are not logically equivalent to the Boolean NK model.

Figure 9. Try6-100

needs to be redone in STELLA

 

 

5.2 What STELLA tells us about the NK model

a. can we get cyclic states in STELLA that are the same as NK

b. if we tweak STELLA with sigmoids and pools do we get different groups of outputs

c. maybe we should consider the increase or decrease in the flows rather than the concentrations (at what point is a component a rate, i.e. can we consider the stocks evidence of a rate).

5.3 Limitations and strengths of the verbal model

6. Conclusions

The three modeling approaches used in this analysis are very different. All of them attempt to describe how algal cells adapt to a change in conditions. They all could converge on optimal composition of the cellular components as described by Shuter (197*). Although the numerical, dynamic model seems to be the obvious preference in the literature, we didn't present any criteria for choosing any one over the other. The dynamic models are good for describing one set of observation in a mechanistic way but we don't know whether a slight change in either initial conditions or logic control structure would give the same patterns of response. Using Boolean NK models we can explore how the grammar of the regulation logic leads to patterns in behavior and we can simultaneously explore all initial conditions.

This paper has at least two important results. First is the demonstration that the regulation logic sets can lead to different steady state outcomes (Figure 6). The second result is that the oscillations that are sometime associated with an ill behaved dynamic model are also represented in the Boolean NK systems, i.e. the unstable regions are determined in part by the underlying logic, not simply the initial conditions, choices of parameters, or timestep in the numerical technique.

*** Inefficiency ? ***

7. References

Burmaster - meta model paper

Flake (2001)

Kana et al 1997 balance hypothesis

Kauffman 1994

Reynolds CSR framework

Shuter optimization paper

Tandeau de Marsaac - two component model reference

8. Links

Appendix 1: Listing of the STELLA program with description.

Appendix 2: Example output from the STELLA program.

Appendix 3: Comparison of logics in STELLA