Reaction order polyhedra reveal hidden adaptive regimes

Chapter II: Polyhedral constraints enable holistic analysis of bioregulation

2.3 Reaction order polyhedra reveal hidden adaptive regimes

orders as

Plasmid number invariance: 𝐺∝𝑡⁰_𝐺, (2.13) that the reaction order of𝐺to𝑡_𝐺 should be zero.

Since the plasmid number invariance function here is considered over the time scale of gene expression, the catalysis, or production and degradation of biomolecules, need to be considered. The important dynamics of catalysis here is the production and degradation of the repressor molecule𝑡_𝑅. Again, total is used here because production and degradation causes integer changes in the total number of repressor molecules. The equation for dynamics is

𝑑

𝑑𝑡𝑡_𝑅=𝑘𝑡_𝐺−𝛾𝑡_𝑅, (2.14)

where production is proportional to plasmid number𝑡_𝐺 because the repressor is constitu- tively expressed on the plasmid, and degradation with rate𝛾describe dilution from cell growth. At steady state, we have

Catalysis steady state: 𝑡_𝑅 = 𝑘

𝛾𝑡_𝐺. (2.15)

As a result of this the repressor amount is proportional to plasmid number.

Now we have described everything in the system. There are three elements, (1) the binding regulation, (2) the desired function of plasmid number invariance, and (3) the steady state relation from catalysis dynamics. All together, they form conditions on reaction orders so that plasmid number invariance is achieved at steady state if and only if this condition on reaction orders is satisfied.

The integration of these three elements into constraints on reaction orders is illustrated in (a) of Figure 2.5. Denote 𝑎_𝑅 and 𝑎_𝐺 as the reaction orders of 𝐺 to 𝑡_𝑅 and 𝑡_𝐺. For intuition and convenience of notation, let us denote that reaction orders roughly mean 𝐺has proportional relation𝐺∝ 𝑡^𝑎_𝑅^𝑅𝑡^𝑎_𝐺^𝐺. The constraint from binding regulation is then that the reaction orders (𝑎𝑅, 𝑎𝐺) are bounded in a polyhedral set 𝒫_bind. This reaction order polyhedra is both derived via DDT, as shown in (b) of Figure2.5, and visualized by computer sampling in (d) of Figure2.5.

Then we incorporate the constraint from catalysis steady state. The steady state relation implies𝑡_𝑅∝𝑡_𝐺, i.e. the repressor amount is proportional to the plasmid number. Therefore, we can write its influence on reaction orders by𝐺∝𝑡^𝑎_𝑅^𝑅𝑡^𝑎_𝐺^𝐺 ∝𝑡^𝑎_𝐺^{𝐺+𝑎𝑅}. Lastly, we incorporate the constraint from the requirement of plasmid number invariance, that𝐺∝𝑡⁰_𝐺. We see that this becomes the constraint𝑎_𝐺+𝑎_𝑅= 0in terms of reaction orders. As a result of all

Figure 2.5Holistic analysis of the plasmid number invariance circuit from [98] reveals invariance regimes previous missed.(a)Circuit specifications form constraints on reaction orders. Constraints on reaction orders (𝑎_𝑅, 𝑎_𝐺)of gene𝐺’s to repressor and plasmid number come from all three parts of the system: binding regulation, catalysis or production-degradation of repressor, and the plasmid number invariance function.

Binding network restricts the reaction orders to the reaction order polyhedron𝒫_bind. The steady state of catalysis dynamics requires that the repressor concentration is proportional to the plasmid number, so 𝐺∝ 𝑡^𝑎_𝐺^𝐺𝑡^𝑎_𝑅^𝑅 =𝑡^𝑎_𝐺^{𝐺+𝑎𝑅}. These together with the constraint from the desired plasmid number invariance function results in the constraints on the reaction orders (circled in red).(b)DDT of target species𝐺. The last step of decomposition results in a vertex (circled by orange) and a ray. Both the vertex and the ray satisfy the reaction order constraints, therefore circled in red. The vertex corresponds to the orange region in (c), and the orange dot in (d). The vertex together with the ray, circled red, corresponds to the region above the black line in (c) and the red line’s intersection with the polyhedron in (d). Upper right corner of the DDT lists the binding network, definition of totals, and the steady state expression for the target species𝐺for book-keeping. (c)Variation in𝐺caused by varying plasmid number𝑡𝐺. White means plasmid number invariance. 𝑦-axis is repressor expression strength𝑘. The orange region is the invariance regime known in [98], corresponds to vertex(−1,1)in reaction orders. The white region above the black line is a previously missed invariance regime, corresponds to the ray in reaction orders. Above the black line is concentrations where𝐶_𝐺𝑅dominate in𝑡_𝐺, so ^𝐶𝐺𝑅

𝐺 ≥10. The orange region is from the above and that𝑅dominates𝑡_𝑅by

𝑅

𝐶_𝐺𝑅 ≥10.(d)Reaction order polyhedron of𝐺obtained by computer sampling, with ^𝐺

𝐾 and ^𝑅

𝐾 log-uniformly sampled between10⁻⁶and10⁶with a total of10⁵points. The orange dot corresponds to the(−1,1)vertex in DDT from (b). The red line is the constraint that𝛼_𝐺+𝛼_𝑅= 0from plasmid number invariance and catalysis steady state.

three elements of the system, we have the following necessary and sufficient condition for plasmid number invariance (see (a) of Figure2.5).

Plasmid number invariance at steady state ⇐⇒ 𝑎_𝑅+𝑎_𝐺= 0,(𝑎_𝑅, 𝑎_𝐺)∈ 𝒫_bind. (2.16) Visually represented in (d) of Figure 2.5, this is the intersection between the line of 𝑎_𝑅+𝑎_𝐺 = 0(red) and the polyhedron𝒫_bind. The result is a vertex(−1,1)with a ray in the (−1,1)direction. Looking at the DDT in (c) of Figure2.5, we see that these two together correspond to the dominance condition 𝑡_𝐺 ≈ 𝐶_𝐺𝑅. In other words, most plasmids are bound.

We emphasize that from this holistic analysis based on combining reaction order polyhedra with the functional constraints, this dominance condition𝑡_𝐺≈𝐶_𝐺𝑅that most plasmids are bound is bothnecessary and sufficient. As long as this condition is satisfied, then plasmid number invariance is achieved. Conversely, if plasmid number invariance is achieved, then this condition is for sure satisfied. This conclusion can only be broken if the system specification is wrong. Specifically, either the repressor amount is not proportional to plasmid number at steady state, or the binding network is incorrect that there are other binding reactions involved.

Now we relate to the analysis in the original paper [98] where this design was proposed and implemented in bacteria. Their analysis is based on Hill functions and Michaelis-Menten type assumptions. As a result, they find the (−1,1) vertex as a functional regime for plasmid number invariance (circled by orange in (b) and the red dot in (d) of Figure2.5).

However, this is a regime contained in the more general conditions necessary for plasmid number invariance, as seen in the DDT (see (c) of Figure2.5). Specifically, the vertex(−1,1) requires two dominance conditions,𝑡𝐺≈𝐶𝐺𝑅and𝑡𝑅 ≈𝑅. In other words, it requires that both most plasmids are bound, and repressor is overabundant so that most repressors are free. The regime missed by this analysis is the ray towards(−1,1)also contained in the constraint in Eqn (2.16). This hidden regime corresponds to dominance conditions that most plasmids are bound with repressors, but most repressors are bound as well. In other words, the regime where repressors bind tightly with the gene on plasmids, while the repressor amount is low. This means, in addition to the repressor-over-expressed scenario considered in [98] to achieve plasmid number invariance, we could use a low expression for the repressor, as long as the repressor is tight.

To show what the hidden regime looks like when concentrations of gene and repressors are varied, we plotted (c) in Figure2.5. The invariance regime that is missed previously is

indeed scenarios where the number of repressors expressed per plasmid is low, while the binding is tight.

All this together, we have shown via one example how the application of holistic bioregula- tion analysis based on reaction order polyhedra can be fruitful in biocircuit design. The power of holistic analysis fundamentally come from its capability to derive necessary and sufficient conditions, rather than just sufficient ones. This allows it to state all possible scenarios a desired function can be achieved. Although this capability is only used here to find a hidden regime, its power in bioengineering is much more profound. Debugging when a design is not performing as desired is part of the foundation for design-build-test cycles in engineering. With only sufficient conditions, debugging can only be done with trial and error. With necessary and sufficient condition for a function in a given circuit design, we can either find how to improve, or conclude that the current circuit design have to be modified.