Initiation of differential gene expression in sporulating Bacillus subtilis – a sceptical biochemist looks at mathematical modelling Michael Yudkin Kellogg College Oxford

Part I. The biological system

Sporulation in Bacillus subtilis Vegetative cycle Sporulation cycle

FF EE Establishing differential gene expression

Three proteins act to regulate  F : SpoIIAB (AB), a homodimer of 33 kDa; SpoIIAA (AA), a monomer of 13 kDa; SpoIIE (IIE), a multidomain protein of 91 kDa.

AB can engage in three interactions: It can make a complex with  F in the presence of ATP; It can make a complex with AA in the presence of ADP; It can use ATP to phosphorylate AA on Ser-58, yielding AA-P. AA is the substrate of the AB kinase. IIE hydrolyses AA-P back to AA.

Default situation – pre-divisional cell and mother cell  F is in a  F ·AB·ATP complex, with one molecule of  F bound to an AB dimer ; AA is phosphorylated to AA-P, which cannot interact with AB ; IIE is absent or almost absent.

Release of  F in the prespore IIE is made and hydrolyses AA-P to AA. The AA attacks the  F ·AB·ATP complex.  F is liberated (“induced release”). AA is phosphorylated to AA-P; AB·ATP is converted to AB·ADP.

Cartoon of induced release FF AB AA AB AA AB AA FF FF

Induced release and its consequences AA +  F ·AB·ATP →  F + AA·AB·ATP AA·AB·ATP → AA-P + AB·ADP AB·ADP  +  ATP → AB·ATP + ADP  F + AB·ATP →  F ·AB·ATP

How is the activation of  F maintained?

AA AA-P AB ATP ADP IIE PiPi H2OH2O Summary: ATP + H 2 O → AA + P i A wasteful cycle? Cycling of SpoIIAA Can the prespore slow down the cycle to diminish the waste of ATP?

Time course of phosphorylation of SpoIIAA

The time course shows that, after one round of phosphorylation, the enzyme has to return from an inactive to an active form. This return clearly includes a slow step. Our experiments have identified the slow step as the loss of ADP from AB·ADP. But we know from previous work that the loss of ADP from AB·ADP is usually extremely rapid. Why is this case different?

Effect of adding ADP on the phosphorylation of AA

We can account for the results by suggesting that an interaction with AA changes the conformation of AB to AB*. AB* binds exceptionally tightly to ADP. AB*·ADP loses its ADP into the solution extremely slowly, and so is long-lived. Unlike AB·ADP, AB*·ADP can interact with AA to make AA·AB*·ADP. The formation of AA·AB*·ADP dramatically slows the phosphorylation of AA.

Conclusions so far While phosphorylating AA, AB accumulates in the form AA·AB*·ADP. The formation of this complex greatly slows the phosphorylation reaction. So long as AB is phosphorylating AA, it cannot interact with  F. Prolonging the phosphorylation reaction therefore serves to maintain the activity of  F.

Phosphorylation reaction scheme AB·ADPAA·AB*·ATP AB·ATP AB*·ADP AA-P AA-P·AB*·ADP AA AB AA AA·AB*·ADP 1 sec -1 ADP ATP

Structure of SpoIIAB

Conformational change in AB AB·ATP AA·AB*·ADP We suggest that the conformational change is a closing of the ATP-lid when AA binds

1 sec -1 ADP ATP

Induced release of  F from the  F ·AB·ATP complex depends on dephosphorylation of AA-P by IIE. Maintenance of free  F over the necessary period also depends on IIE activity. Where is the IIE activity located? Source of dephosphorylated AA

 II E IIE is a membrane-bound enzyme, found at the asymmetric septum. Is IIE activity confined to the prespore? If so, how?

Part II. The mathematical model

Mathematical models (a sceptical biochemist’s view) How do they work? What are they for? Why are biochemists sceptical of them? How do we know if our model is any good?

How do models work? We translate on and off rates of individual interactions into a set of equations, e.g. A B C k1k1 k2k2 k −1 k −2 k3k3 k−3k−3 = k 1 ·B + k −2 ·C − k −1 ·A − k 2 ·A dAdA dt

The kinetic constants (k 1, k 2 etc.) and the concentrations of the components ([A], [B] etc.) are the parameters. These parameters are used to produce a set of linked differential equations. These linked equations constitute the model.

What are models for? I: Qualitative aspects A mathematical model should include all the components of a system and all of their interactions. If the model fails to simulate the known behaviour of the system, it is likely that one (or more than one) of the interactions has been omitted.

What are models for? II: Quantitative aspects Genetics and biochemistry identify the components in a system and show how these components interact. But to get a quantitative view of the outcome of such interactions, a verbal description is not enough.

The need for a quantitative view: IIE activity in the prespore As we saw earlier, IIE activity is located on the asymmetric septum. The prespore is ~ 5-fold smaller in volume than the mother cell. So the effective concentration of IIE is 5-fold higher in the prespore. Does this difference account for the specificity of gene expression?

The need for a quantitative view: the  F /  A paradox In the prespore  F has to displace  A from the core RNA polymerase. But the affinity of core polymerase for  F is 25-fold less than for  A, and the concentration of  F is only 2-fold higher. So how can  F displace  A ?

Why are biochemists sceptical of mathematical models? The number of interactions involved in a regulatory scheme is often large. If we want to make the mathematics come out right, we may be tempted: to change the parameters without regard to whether they make physical sense; to increase the number of interactions without regard to the principle of parsimony.

How do we know if our model is any good? The values for all the parameters should be justifiable (because they have been measured explicitly, or because they are similar to those measured in analogous systems) ; The model should make predictions that we can test experimentally.

The  F regulatory interactions AB + ATP → AB·ATP  F + AB·ATP →  F ·AB·ATP AA +  F ·AB·ATP →  F + AA·AB*·ATP AA·AB*·ATP → AA-P + AB*·ADP AA-P + H 2 O → AA + P i AA + AB*·ADP → AA·AB*·ADP AB·ADP  +  ATP → AB·ATP + ADP  F + core RNAP →  F ·holoenzyme

We have previously measured the kinetic constants for these interactions. We have also measured the concentrations of the intermediates. We can make a plausible estimate of the kinetic constants for the conformational changes in AB. We thus have a set of parameters, with which we can construct a model – Model 1. We now use Model 1 to make verifiable predictions, both qualitative and quantitative.

Predicted formation of  F -holoenzyme (Model 1)

Evidently some factor is missing from our model. What could it be? AB is a dimer, and we know that it can undergo conformational changes; so maybe AB is an allosteric protein. If AB is allosteric, it is possible that AA binds to it cooperatively. Cooperativity can now be included in the model, to give Model 2.

Sporulation Model 2

Predicted formation of  F -holoenzyme (Models 1 and 2)

Binding of AA to AB – experiment and simulation

Other simulations from Model 2 The model can successfully simulate results obtained in vitro from experiments on: Binding of AA to AB·ADP Binding of  F to AB·ATP Disruption of  F ·AB·ATP complexes by AA Rebinding of  F to AB·ATP as AA is phosphorylated Response of this rebinding to IIE Time course of phosphorylation of AA.

Quantitative aspects of Model 2: IIE activity in the prespore As we saw earlier, IIE activity is located on the asymmetric septum. The prespore is ~ 5-fold smaller in volume than the mother cell. So the effective concentration of IIE is 5-fold higher in the prespore. Does this difference account for the specificity of gene expression?

Predicted release of  F

IIE+AA-P IIE

Quantitative aspects of Model 2: the  F /  A paradox In the prespore  F has to displace  A from the core RNA polymerase. But the affinity of core polymerase for  F is 25-fold less than for  A, and the concentration of  F is only 2-fold higher. So how can  F displace  A ?

Predicted formation of  A -holoenzyme and  F -holoenzyme

Conclusions of a convinced sceptic When used judiciously and critically, mathematical modelling is an extremely valuable technique. Mathematical modelling can supply insights that could not be reached by any other method now available.

Acknowledgments The model was developed by Joanna Clarkson and Dagmar Iber, using data obtained in my lab by the following: Kyung-Tai Min Mahmoud Najafi Thierry Magnin Matt Lord Daniela Barillà Brian Lee Isabelle Lucet Jwu-Ching Shu Joanna Clarkson