Glyolysis is a way that organisms can get free energy from glucose without needing oxygen. Animals like you and me can do glycolysis, but we get more free energy from oxidizing glucose. Other organisms are anaerobic: they don’t need oxygen. And some, like yeast, survive mainly by doing glycolysis!
If you put yeast cells in water containing a constant low concentration of glucose, they convert it into alcohol at a constant rate. But if you increase the concentration of glucose something funny happens. The alcohol output starts to oscillate!
It’s not that the yeast is doing something clever and complicated. If you break down the yeast cells, killing them, this effect still happens. People think these oscillations are inherent to the chemical reactions in glycolysis.
I learned this after writing Part 1, thanks to Alan Rendall. I first met Alan when we were both working on quantum gravity. But last summer I met him in Copenhagen, where we both attending the workshop Trends in reaction network theory. It turned out that now he’s deep into the mathematics of biochemistry, especially chemical oscillations! He has a blog, and he’s written some great articles on glycolysis:
• Alan Rendall, Albert Goldbeter and glycolytic oscillations, Hydrobates, 21 January 2012.
• Alan Rendall, The Higgins–Selkov oscillator, Hydrobates, 14 May 2014.
In case you’re wondering, Hydrobates is the name of a kind of sea bird, the storm petrel. Alan is fond of sea birds. Since the ultimate goal of my work is to help our relationship with nature, this post is dedicated to the storm petrel:
Last time I gave a summary description of glycolysis:
2 pyruvate + 2 NADH + 2 H+ + 2 ATP + 2 H2O
The idea is that a single molecule of glucose:
gets split into two molecules of pyruvate:
The free energy released from this process is used to take two molecules of adenosine diphosphate or ADP:
and attach to each one phosphate group, typically found as phosphoric acid:
thus producing two molecules of adenosine triphosphate or ATP:
along with 2 molecules of water.
But in the process, something else happens too! 2 molecules of nicotinamide adenine dinucleotide NAD get reduced. That is, they change from the oxidized form called NAD+:
to the reduced form called NADH, along with two protons: that is, 2 H+.
Puzzle 1. Why does NAD+ have a little plus sign on it, despite the two O–’s in the picture above?
Left alone in water, ATP spontaneously converts back to ADP and phosphate:
This process gives off 30.5 kilojoules of energy per mole. The cell harnesses this to do useful work by coupling this reaction to others. Thus, ATP serves as ‘energy currency’, and making it is the main point of glycolysis.
The cell can also use NADH to do interesting things. It generally has more free energy than NAD+, so it can power things while turning back into NAD+. Just how much more free energy it has depends a lot on conditions in the cell: for example, on the pH.
Puzzle 2. There is often roughly 700 times as much NAD+ as NADH in the cytoplasm of mammals. In these conditions, what is the free energy difference between NAD+ and NADH? I think this is something you’re supposed to be able to figure out.
Nothing in what I’ve said so far gives any clue about why glycolysis might exhibit oscillations. So, we have to dig deeper.
Glycolysis actually consists of 10 steps, each mediated by its own enzyme. Click on this picture to see all these steps:
If your eyes tend to glaze over when looking at this, don’t feel bad—so do mine. There’s a lot of information here. But if you look carefully, you’ll see that the 1st and 3rd stages of glycolysis actually convert 2 ATP’s to ADP, while the 7th and 10th convert 4 ADP’s to ATP. So, the early steps require free energy, while the later ones double this investment. As the saying goes, “it takes money to make money”.
This nuance makes it clear that if a cell starts with no ATP, it won’t be able to make ATP by glycolysis. And if has just a small amount of ATP, it won’t be very good at making it this way.
In short, this affects the dynamics in an important way. But I don’t see how it could explain oscillations in how much ATP is manufactured from a constant supply of glucose!
We can look up the free energy changes for each of the 10 reactions in glycolysis. Here they are, named by the enzymes involved:
I got this from here:
• Leslie Frost, Glycolysis.
I think these are her notes on Chapter 14 of Voet, Voet, and Pratt’s Fundamentals of Biochemistry. But again, I don’t think these explain the oscillations. So we have to look elsewhere.
By some careful detective work—by replacing the input of glucose by an input of each of the intermediate products—biochemists figured out which step causes the oscillations. It’s the 3rd step, where fructose-6-phosphate is converted into fructose-1,6-bisphosphate, powered by the conversion of ATP into ADP. The enzyme responsible for this step is called phosphofructokinase or PFK. And it turns out that PFK works better when there is ADP around!
In short, the reaction network shown above is incomplete: ADP catalyzes its own formation in the 3rd step.
How does this lead to oscillations? The Higgins–Selkov model is a scenario for how it might happen. I’ll explain this model, offering no evidence that it’s correct. And then I’ll take you to a website where you can see this model in action!
Suppose that fructose-6-phosphate is being produced at a constant rate. And suppose there’s some other reaction, which we haven’t mentioned yet, that uses up ADP at a constant rate. Suppose also that it takes two ADP’s to catalyze the 3rd step. So, we have these reactions:
fructose-6-phosphate + 2 ADP → 3 ADP
Here the blanks mean ‘nothing’, or more precisely ‘we don’t care’. The fructose-6-biphosphate is coming in from somewhere, but we don’t care where it’s coming from. The ADP is going away, but we don’t care where. We’re also ignoring the ATP that’s required for the second reaction, and the fructose-1,6-bisphosphate that’s produced by this reaction. All these features are irrelevant to the Higgins–Selkov model.
Now suppose there’s initially a lot of ADP around. Then the fructose-6-phosphate will quickly be used up, creating even more ADP. So we get even more ADP!
But as this goes on, the amount of fructose-6-phosphate sitting around will drop. So, eventually the production of ADP will drop. Thus, since we’re positing a reaction that uses up ADP at a constant rate, the amount of ADP will start to drop.
Eventually there will be very little ADP. Then it will be very hard for fructose-6-phosphate to get used up. So, the amount of fructose-6-phosphate will start to build up!
Of course, whatever ADP is still left will help use up this fructose-6-phosphate and turn it into ADP. This will increase the amount of ADP. So eventually we will have a lot of ADP again.
We’re back where we started. And so, we’ve got a cycle!
Of course, this story doesn’t prove anything. We should really take our chemical reaction network and translate it into some differential equations for the amount of fructose-6-phosphate and the amount of ADP. In the Higgins–Selkov model people sometimes write just ‘S’ for fructose-6-phosphate and ‘P’ for ADP. (In case you’re wondering, S stands for ‘substrate’ and P stands for ‘product’.) So, our chemical reaction network becomes
S + 2P → 3P
and using the law of mass action we get these equations:
where and stand for how much S and P we have, respectively, and are some constants.
Now we can solve these differential equations and see if we get oscillations. The answer depends on the constants and also perhaps the initial conditions.
To see what actually happens, try this website:
• Mike Martin, Glycolytic oscillations: the Higgins–Selkov model.
If you run it with the constants and initial conditions given to you, you’ll get oscillations. You’ll also get this vector field on the plane, showing how the system evolves in time:
This is called a phase portrait, and its a standard tool for studying first-order differential equations where two variables depend on time.
This particular phase portrait shows an unstable fixed point and a limit cycle. That’s jargon for saying that in these conditions, the system will tend to oscillate. But if you adjust the constants, the limit cycle will go away! The appearance or disappearance of a limit cycle like this is called a Hopf bifurcation.
For details, see:
• Alan Rendall, Dynamical systems, Chapter 11: Oscillations.
He shows that the Higgins–Selkov model has a unique stationary solution (i.e. fixed point), which he describes. By linearizing it, he finds that this fixed point is stable when (the inflow of S) is less than a certain value, and unstable when it exceeds that value.
In the unstable case, if the solutions are all bounded as there must be a periodic solution. In the course notes he shows this for a simpler model of glycolysis, the Schnakenberg model. In some separate notes he shows it for the Higgins–Selkov model, at least for certain values of the parameters:
• Alan Rendall, The Higgins–Selkov oscillator.