Team:Edinburgh/Cellulases (Kappa model)

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Cellulases (Kappa model)

Kappa

Kappa is a stochastic, agent-based language for simulating interactions in (mostly) biological systems. It has been used by several iGEM teams before, most notably by Team Edinburgh 2010, who used it to develop a model awarded the Best Model Prize.

A model in Kappa would be represented in terms of agents and rules. Agents could be thought of as 'idealised proteins' (Danos et al., 2008), or other entities (like cells). They can have multiple sites, which can bind to one another; and states of different values. For example, an agent called Glucose in our system is defined as:

%agent: Glucose(end1, end4, belongs~cellulose~cellobiose~nothing~complex)

That means that a molecule of glucose has two sites: end 1, end 4 (meaning 1' and 4' carbons) and one state: belongs (which is only really used for the purpose of counting different substances).

Reactions define the interactions between the agents. The left side of the reaction defines the preconditions of reaction, and the right side represents what happens to the substrates. A reaction can create or delete bonds between chemicals, create or delete agents, and change states of agents. A reaction also happens at a certain rate. For example, cutting cellulose in the middle of the chain by Endoglucanase can be represented as the following rule:

Endoglucanase(activeSite),
Glucose(end1?, end4!1, belongs~cellulose),
Glucose(end1!1, end4!2, belongs~cellulose),
Glucose(end1!2, end4!3, belongs~cellulose),
Glucose(end1!3, end4?, belongs~cellulose) ->
                                Endoglucanase(activeSite),
                                Glucose(end1?, end4!1, belongs~cellulose),
                                Glucose(end1!1, end4, belongs~cellulose),
                                Glucose(end1, end4!3, belongs~cellulose),
                                Glucose(end1!3, end4?, belongs~cellulose)
                                @ 'γ_Endo_forward'

This means that an agent called Endoglucanase with a free active site (i.e. not bound to an inhibiting molecule) needs to be present with a chain of at least 4 glucoses, and as a result one of the bonds within cellulose chain is cut. This happens at a rate 'γ_Endo_forward', which is a defined constant.

The Kappa simulator uses 'biological time', which is roughly corresponding to actual 'wall clock time'. The state of the system is evolved by choosing a reaction to happen and a corresponding time step to take. These are selected in a stochastic way, based on the following factors:

The kinetic rates of reactions contribute towards the 'activity' of every reaction rule. More active rules get higher probability of firing next.
The time step is also dependent on activity of rules.

Structure of the model

We thought that a neat model would represent cellulose as a long fibre of glucoses linked together:

Cellulose represented as long fibres of glucose

However, manually inputting that would be a considerable effort. Instead, we made a Kappa system which would build cellulose for us, and then, using Kappa syntax, took a snapshot of the model. Two variants were created, one with long (10,000 glucoses) fibres of cellulose and another one with short (1,000 glucoses) fibres.

Then, Endoglucanase would cut Cellulose somewhere in the middle:

Endoglucanase digests cellulose chains

Also, Exoglucanase would cut Cellulose at end-points:

Exoglucanase digests cellulose chains

Fontes and Gilbert (2010) have suggested that Exoglucanase is inhibited by its end product - Cellobiose, so we modelled that interaction as well.

Finally, β-glucosidase would cut dimers (i.e. Cellobiose):

β-glucosidase digests cellobiose

We also modelled end-product inhibition of β-glucosidase by glucose.

Non-synergistic model

We developed our first, non-synergistic model in Kappa, based on the system as seen above. We carried out a series of experiments in order to fit parameters to our model. The aim was to generate graphs similar to those in Kadam, Rydholm & McMillan (2004) (since back then we did not have any data available from the wet lab). We manipulated the kinetic rates of reactions and relative concentrations of enzymes.

After an overnight series of experiments using a computing cluster in the School of Informatics, we came up with a set of values for parameters of our model. We would stick to these values of parameters across all our Kappa models.

Dependency of model on kinetic rates

During this experimental phase, we found that the model dynamics is hugely affected by varying the reactions' kinetic rates. The following graphs present the effect of varying single kinetic rates, with all other being equal. We tested combinations of all the reaction rates, but these are omitted from the graphs for clarity.

The effect of manipulating reaction rates on cellulose degradation rate.
Left column represents the kinetic rate of parameter in question at 1·10^- 8 (less reactivity),
right column :1·10^- 2(more reactivity)

The activity of the three enzymes influences the behaviour of the system in slightly different ways.

β-glucosidase influences how quickly cellobiose is degraded into glucose
endoglucanase influences the level of partial degradation of cellulose
exoglucanase influences the amount cellobiose available to β-glucosidase.

The effect of β-glucosidase and exoglucanase activity is as expected and logically following from how the reactions are defined. higher activity of these enzymes causes faster cellulose degradation.

High activity of endoglucanase was found to place a (temporary) limit on production of glucose.

The effect of manipulating inhibition rates on cellulose degradation rate.
Left column represents the inhibition rate of parameter in question at 1·10^- 8 (weaker inhibition),
right column :1·10^- 2(stronger inhibition)

Increase in all the inihibition rates causes a decline in cellulose degradation. β-glucosidase inhibition is has the most influence on the time taken to fully degrade cellulose, because the enzyme inhibited is the one producing the end product of the reaction chain.

Synergistic model

Since Kappa models can be easily extended (by adding new agents, or reaction rules), we tried to make a synergistic model based on the non-synergistic model.

We introduced new agents: an E.Coli cell and several copies of INP, to be linked together as shown below:

A diagram of the synergistic (yet non-spatial) model.

However, that did not act as we predicted. The following graph presents both models.

A graph showing the output of both synergy and non-synergy models. Dotted lines indicate the non-synergistic model

The graph above shows that for the same set of parameters, there is virtually no difference between the two systems. This is because Kappa does not have the notion of proximity, i.e. the simulator has no way of distinguishing between the products of enzymes produced just a second ago and the ones floating far away in the solution:

A diagram showing two cellobiose molecules, one close to the enzymes, and one far away in the solution. However, Kappa simulator does not distinguish between them.

You can download the ready model as a .zip file (instructions in README file):

Spatial Kappa

Spatial Kappa is an extension to Kappa made by our last year's team (Team Edinburgh 2010). It seems like the extra functionality of compartments added by Spatial Kappa would help develop a model of synergistic action.

Our model could work in the following way:

In the synergistic system, the enzymes are all in one compartment. The products and substrates would be allowed to diffuse across compartments, while the enzymes would not.
The non-synergistic system would be very similar, but the enzymes could diffuse across compartments.

The resulting system should show that enzymes being in close proximity degrade cellulose faster.

Because of the design choice we made at the beginning - representing cellulose as a chain of glucoses - we encountered a problem while setting up diffusion rules. It is trivial to set up diffusion rules for enzymes (just one rule per enzyme), but for the diffusion of cellulose, we would need to set up a separate rule for every possible length of cellulose chain - in our case that would amount to 1000 rules. While it is possible to automatically generate the rules (e.g. using Python), this would slow the model down, meaning it would become intractable.

Therefore we tried a different representation of cellulose, where the agent Cellulose has a length n,

n ∈ {1..10}

(representing a chain of cellulose of length 2ⁿ⁺¹ glucoses).Then,

Endoglucanase would take a chain of length n, and produce two chains of length n-1 (or two cellobioses from a chain of length 1);
Exoglucanase would take a chain of length n, and produce n-1 chains of decreasing length, connected to each other, plus an extra chain of length n = 1, plus a particle of Cellobiose, thus conserving the amount of glucose:

2ⁿ⁺¹ = 2ⁿ + 2^n-1 + … + 2³ + 2² + 2 + 2

Endoglucanase could then split chains made in Exoglucanase reaction;
β-glucosidase would split Cellobiose into two glucoses;
β-glucosidase is inhibited by its end-product, glucose;
Exogluconase is inhibited by cellobiose.

A diagram of the spatial model.
Compartments are represented by squares, the arrows indicate diffusion of relevant chemicals. Chains represent cellulose, cellobiose and glucose.
Left side shows then synergistic model, the non-synergistic model is shown in the right.
Note that the amounts of enzymes and substrates is identical in both models.
You can also view it as an animation.

An example simulation of the system is shown below. You can see the difference between synergistic and non-synergistic system.

A graph showing both synergistic and non-synergistic models of cellulosome.
Note: The steps in the graph are artifacts from the fact that a large molecule of cellulose can enter the compartment with enzymes and therefore suddenly a large amount of cellulose is available to be digested.

TODO: a graph of the basic model why so rough? because when a large molecule of cellulose goes into [1][1], suddenly a lot of substrates is available.

Manipulating the initial mixture

In real life, cellulose might exist in the form of a mixture of chains of varying length. As a result of the way we represented cellulose in Spatial Kappa, it is easy to

Additional cellulose

In a real-world biorefinery, one would prefer to constantly add cellulose to (and remove glucose from) the bioreactor. The point at which you add cellulose to the system can be optimised with respect to glucose output over time.

Kappa can simulate exactly that - adding or removing agents from the reaction mixture. We could test a range of conditions in which cellulose is added, e.g.:

when a certain percent of cellulose is degraded,
at fixed time intervals,
when a certain amount of end-product is produced.

References

Danos V, Feret J, Fontana W, Harmer R, Krivine J (2008) Rule-Based Modelling, Symmetries, Refinements. Lecture Notes in Computer Science 2008(5054): 103-122 (doi: 10.1007/978-3-540-68413-8_8)
Fontes CMGA, Gilbert HJ (2010) Cellulosomes: highly efficient nanomachines designed to deconstruct plant cell wall complex carbohydrates. Annual Review of Biochemistry 79: 655-81 (doi: 10.1146/annurev-biochem-091208-085603).
Kadam KL, Rydholm EC, McMillan JD (2004) Development and validation of a kinetic model for enzymatic saccharification of lignocellulosic biomass. Biotechnology Progress 20(3): 698–705 (doi: 10.1021/bp034316x).

@@ Line 148: / Line 148: @@
 == Manipulating the initial mixture ==
-In real life, cellulose might exist in the form of a mixture of long and short chains. We can model that in Kappa.
+In real life, cellulose might exist in the form of a mixture of chains of varying length. As a result of the way we represented cellulose in Spatial Kappa, it is easy to
 == Additional cellulose ==