## Saturday, March 23, 2013

### Choke points, Load points, Hots spots and Key steps

March 23, 2013 8:56 pm

Choke points, load points, hot spots, key steps and critical steps are some of the many phrases that have been coined to describe places in a biochemical network where perturbations are said to make a difference. Locating these places is important because they are sites where we would target a drug or reengineer to change a cell's phenotype. The question is what are these points, is a choke point something that can be identified as a physical thing, can a choke point be removed from a pathway and studied in isolation? To answer these questions let us look at a very simple two step pathway. Both steps in the pathway are governed by reversible Michaelis-Menten kinetics, so nothing out of the ordinary. Let's say that a perturbation to the enzyme activity at the second step results in a significant change in the steady-state flux through our little pathway, The first step has hardly any effect. We might therefore call the second step the choke point of the pathway, or the rate-limiting step, the hot spot etc.

Is there something special about the second step that makes it a 'choke point'? What may surprise many is that it is the first step that makes the second step a choke point. There is nothing intrinsically special about the second step that makes it the choke point, all the interesting action occurs at the first step.

The explanation is simple. The reaction rate through the first step is very sensitive to the product it makes. If the product of the first step goes up a bit the result is that the reaction rate through the first step goes down, this is simply due to strong product inhibition, that is the product is an effective competitor with respect to the substrate for the first step. So what? Consider this scenario, we add a drug that can inhibit the activity of the second step. Its reaction rate goes down. This results in the substrate for the second step to go up (its being consumed less so it must go up). Since the first step is sensitive to this concentration, its rate goes down. The net effect is that it looks as if the second step is the choke point. The name choke suggests a small diameter pipe so that making the pipe smaller has a marked effect on the flow. But this isn't what is happening here. What is happening is that there is negative feedback from the second step to the first step and it is the first step that slows down the flow. If it weren't for the product inhibition on the second step, the second step would have NO effect on the flow. The second step is actually NOT a choke point at all.

The words we use can be very misleading especially when dealing with complex systems. In this case the work choke is extremely misleading. What about the other phrases that people use, such as a hot spot. Hot spot is certainly a colorful phrase to use but is it a phrase that should be used by intelligent people? Its certainly a word that politicians or my 4 your old might understand, but is it a word that professional scientists should use? What about the word key step or critical step? These don't convey the point that the step itself is not the one responsible for it being influential, it was the first step that gave the second step its ability to control flux. If anything the key step is the first step. These words aren't therefore really appropriate either. What about the word rate-limiting step, still in use today in some circles? That word also suggests a pipe that is too small to carry the flow, it has the same problem as the word choke point.

Most of the words we use are either imprecise or just misleading. What do we use instead? As scientists we should define precisely what we mean, preferably operationally and quantitatively. Forty years ago such a definition was given, that definition is the flux control coefficient, it is simply a number that tells us how much of an effect a perturbation has on the system as a whole. It doesn't explain why a particular step has a given a degree of influence, it just tells us how much influence the step has.

Metabolic pathways such as the amino acid biosynthetic pathways which have allosteric negative feedback are an interesting case. The old literature (and even some new literature) will refer to the step that is regulated by the allosteric effector the rate-limiting step, the choke point or hot spot in more recent language. But modifying this step actually has little effect on the phenotype. It turns out that its the step after the feedback loop that has most of the influence. However it is the regulated step that allows influence to be on the last step. The allosterically regulated step is an important step but it certainly isn't the rate-limiting step, choke point or hot spot.

A real choke point, The Battle of Thermopylae.
Leonidas at Thermopylae, by Jacques-Louis David, 1814.
Image and data provided by Columbia University.

## Sunday, March 17, 2013

### Irreversible Bistability

March 17, 2013 8:40 pm

Last week I was in need of an irreversible bistable switch to show my class. I remember seeing one in the well-known review by Tyson et al, Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cells, Current Opinion in Cell Biology 2003, 15:221–231. I hunted down the paper and found it. However, when I looked at the details of the model, it turns out to be incorrect or at least very odd. I couldn't get the model to work and I didn't particularly like the equations that described the model (assuming they were correct and not a typo in the paper). So I decided to design my own. The traditional bistable system has two physically realizable steady states, that is states that admit positive concentrations. By suitably large perturbations it is possible to jump from one state to the other. We often depict bistable systems in the form of a bifurcation diagram as shown below:

Here we see the state of the system (vertical axis) plotted as a function of a parameter in the system. Over a particular range of the parameter, we see that the system admits three possible solutions (along a vertical line). Two are stable and the third is unstable (blue dotted line). If I start on at a high steady-state at a parameter value of say 1.2 and slowly increase the parameter, the steady-state value will slowly decrease until I reach a parameter value of about 2.4 where we suddenly jump to the lower steady state. If I now decrease the parameter value I now follow the lower steady-state branch until I reach about 1.3 where I will suddenly jump up to the high state. I can therefore move freely between the high state and the low state simply by changing the parameter. The bistable system is therefore freely reversible.It is possible however to arrange things so that the entire curve is shifted to the left in such a way that is becomes impossible to freely move back and forth between the states.

In the above diagram, we see that the left-hand turning point now pokes into the negative half of the graph. The graph is mirror image of the previous one because I used a slightly different model in the upper one, but don't let that worry you. The negative half implies negative concentrations which is clearly impossible. Consider then starting the parameter (this time called the signal) at zero with the variable also at zero and slowly increasing the signal level. The output concentration will slowly increase following the lower branch until we reach a signal value of about 2.0 when we suddenly jump to the high state. Let us now start lowering the signal. As we lower the signal we now follow the upper steady state branch but instead of dropping to the lower state we end up stuck at the upper branch as we reach a signal strength of zero (My signal can't go below zero concentration!). We are therefore unable to drop to the lower branch. This is what we mean by an irreversible bistable systems. In one direction we can go from the lower steady-state to the upper one, but we cannot go the other way.

To design this system I knew that I could probably start with Tyson's original figure which I redraw below:

See textbook

p = defn cell
$X -> R1; k1*EP + k2*Signal; R1 ->$w; k3*R1;
EP -> E; Vm1*EP/(Km + EP);
E -> EP; ((Vm2+R1)*E)/(Km + E);
end;
p.Vm1 = 12; p.Vm2 = 6;
p.Km = 0.6;
p.k1 = 1.6; p.k2 = 4;
p.E = 5; p.EP = 15;
p.k3 = 3; p.Signal = 0.1;

The \$ sign in front of a species means that it is fixed in time. The first question is what values for the parameters will yield bistabilty? To do that I set up two loops. The outer loop randomized parameters, the inner loop randomized the initial conditions for the system and computed the steady-state. The inner loop ran 10 times and printed out ten steady-state levels of R, the output species. The outer loop ran 50 times. If any of the outer loop parameter combinations generated a bistable system I would see at least to different output levels in R when the inner loop was run. In this way, I was able to quickly locate a parameter set that gave me a bistable system. This was all programed in Jarnac.

Once I had a bistable system I then needed to compute the bifurcation plot shown above. To do this I used Oscill8 (Developed in Tyson's lab), a reliable application for doing simple bifurcation plots and much easier to use than XPP. Oscill8 is SBW compatible so it shows up in the Jarnac menu and all I had to do was call it from Jarnac and the model would be automatically passed over. Alternatively, I could have saved the model from Jarnac in SBML format and loaded it manually into Oscill8 (it also understands SBML). Oscill8 confirmed for me that I did indeed have a bistable system. To get an irreversible system I then used Oscill8 to test every parameter in the system to find the parameter that moved the bifurcation graph left or right. Once I found the right parameter I just moved the bifurcation plot to the left so that the upper loop poked into the negative half of the graph. And that's how I designed my irreversible bistable system to show class. The Jarnac script shown above is the set of parameters that gives the irreversible bistable system.