• coming soon

• ProMot Source Code
• MatLab Source Code
  • hcm_v4.m
  • hcm_ode.m

• used files:
  • batch.do for batch simulation in DIVA
  • plotbatch.m for plotting DIVA and MatLab simulation and experimental data
• The model shows very good agreement with experimental data. (because it was fitted to the data)
• DIVA and Matlab show same quality of simulation.
• DIVA will be used for further continuous studies.

• conti_200_015.do (FRU_0=20.0g/L & AMC_0=1.5g/L)
• plotconti.m (Matlab)
• plotconti4paper.m (Matlab)

plotconti.m generates to figures (fig1, fig2), which were combinded by using xfig to the follwoing figure:

New matlab file plotconti4paper.m gives a better plot (FRU_in=20.0g/L, AMC_in=1.5g/L):

• Three bifurcation points were detected:
  • A: trans critical bifurcation, washout point
  • B: turning point
  • C: continuation fails at this point, predictor stepsize became too small
• Washout point (A) is in very good agreement with data from literature (maximum growth rate mu = 0.31 1/h)

• continuations for different AMC concentrations were done. FRU was constant at 20.0g/L (black curves)
  • 0.1 g/L AMC: conti_200_001.do
  • 0.5 g/L AMC: conti_200_005.do
  • 1.5 g/L AMC: conti_200_015.do
  • 2.5 g/L AMC: conti_200_025.do
  • 3.4 g/L AMC: conti_200_034.do
  • 4.0 g/L AMC: conti_200_040.do
• Width of hysteresis did not change much, but height
• Continuation of two parameters: AMC and D, limit point continuation (red curve), limcon2.do
• Two parameter continuation of bifurcation point C was not possible, limcon1.do
• Plotting was done with MatLab, plotconti_diff_feed.m

• To check, whether concentration of FRU has influence at the hysteresis, FRU was now set to 10g/L and AMC was changed similary:
  • 0.1 g/L AMC: conti_100_001.do
  • 0.5 g/L AMC: conti_100_005.do
  • 1.5 g/L AMC: conti_100_015.do
  • 2.5 g/L AMC: conti_100_025.do
  • 2 par continuation: limcon2fru10.do
  • Plots with MatLab: plotconti_diff_feed2.m
• Results are similar. No influence of FRU was observed. See following two pictures:

It seems, that only the fraction of FRU/AMC is important, not the absolut values. However, for very low concentrations the saturation constants will become more important and therefore results might differ.

Limitpoint continuation of point C was done stepwise: AMC_in was changed and D of point C was found graphically.

Gamma = AMC_in/FRU_in.

As seen in the figure, the width of bifurcation remains almost constant over a wide range of gamma. Both bifurcation point vanishes for gamma > 0.17, e.g. AMC_in concentration of 17%. Height of bifurcation changes. Bifurcation unfolds from gamma = 0 and flattens more and more for higher gamma until it vanishes.

• Continuation in point C fails, since this point is discontinuous.
• This is due to switching between model variants in this point triggered by the cybernetic variable v.

• Model switching occurs because the rates rM_1 (blue) and rM_2 (red) change their importance.
• For low dilution rates rM_1 is the maximum rate and for high dilution rates rM_2 is the maximum rate

For further studies the model (ProMot file) was changed in that way, that either the ROI pv1 or pv2 were fixed as maximum ROI. However, this won't be true for the whole dilution rate interval, but would give some insights. To do so, just the equation for pvmax was changed in ProMot Source Code to

   ("pvmax"
     :system-theoretic "help"
     :value "pv[1]"
     ) 
 

and

   ("pvmax"
     :system-theoretic "help"
     :value "pv[2]"
     ) 
 

respectivly and these models were saved as hcm1a.mdl and hcm2a.mdl respectivly.

For both model variants continuous simulations were done, as seen in the following figure (matlab: comp_model_vars.m). Blue curve represents model 1 (pvmax=pv1) and red curve represents model 2 (pvmax=pv2):

In this figures one can see, that the bifurcation point C is located were both model variants cross each other. However, these model variants are not valid for the whole dilution rates. model variant 1 is only valid if pv1>=pv2 and variant 2 only if pv1⇐pv2. At the bifurcation point C both model variants are valid, since in this point pv1=pv2.

It is interesting that a 5 times higher PHB process yield can be achieved with such a control law (see figure). Hence the first idea one can get is to influence the metabolic control system in a similar way. But the higher PHB yield exists only in the non-valid part of the model. In this part the metabolic control variable will exceed 1 (v>1), which leads to a higher rate in the model (r>r_max). But the real biological system might not be able to achieve such a rate, even by overexpressing of genes.

So, I conclude: this is a nice model to show fancy bifurcations, but probably useless for metabolic engineering.

Most models in literature focus only on PHB synthesis, but neglect PHB degradation. As own (previews) studies have shown, PHB degradation might be crucial for overall dynamic behavior. In the following PHB degradation is removed from model by setting kr_5 = 0.0.

Without PHB degradation no bifurcations points could be detected (except washout point). This is interesting, since as shown before the bifurcation results from the switching between rM1 and rM2, but degradation rate is rM5.

Explanation: As seen in the figure there is still this sharp edge, which was bifurcation point C before. However stability doesn't change anymore in this point. This sharp edge is due to switching between two model variants as shown before. And these model variants do the same things as before, even without degradation. Therefore there is still this sharp edge. But bifurcation point B isn't there anymore. This has the following reason: Bifurcations are often observed in system with two or more substrates. In this system there are two substrates namely FRU and AMC, but these are not substitutable. And we have only one carbon source substrate, namely FRU. However, since we have introduced the dynamics of PHB in the model, PHB became product and substrate, therefore there exist a region now, where there are two carbon substrates namely FRU and PHB. However, PHB is internal and cannot be controlled like the FRU concentration. The organism by itself controls it and will uptake PHB or not. And PHB is uptaken at low AMC concentrations and low dilution rates. At high dilution rates PHB is always produced and from y>0.17 (high AMC concentration). At a first glance that looks weird, but this is still reasonable. At low AMC concentration the organism has excess carbon source and will therefore synthesis PHB, but PHB is also utilized. That means the organism is wasting external carbon source. This might not sound reasonable but actually is. Because the organism can not grow faster since growth is limited by AMC, so the organism will waste excess carbon in order that other bacterias have less carbon available. This is also a kind of strategy to survive and eliminate competitors. On the other hand at low AMC concentration but high dilution rates FRU becomes the limiting substrate and the organism can not waste it anymore. At high AMC concentration, AMC is not limiting growth anymore and the organism won't waste fructose. That is why, we have this bifurcation at low AMC concentration and low dilution rates, since there are two carbon substrates. If degradation not included in the model, then there is always only one carbon substrate and therefore no bifurcation is observed.