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Model

Model for Dockerin-Cohesin interaction

Background

This model aims to investigate the binding of cohesin and dockerin used to link our cellulases and boosters to the scaffold. In earlier stages of our experiment, we wanted to fuse dockerin I and RFP to better monitor the cohesin-dockerin interaction but we weren't very successful, this meant we couldn't visualize the cohesin-dockerin binding if it was accomplished. To ensure that our design of the cellulose-degrading complex is valid, we turned to modelling cohesin-dockerin interactions as an alternative to our experimental verification.

PCR cloning on June 25th; All RFP has no dockerin I successfully fused on it
PCR cloning on June 25th; All RFP has no dockerin I successfully fused on it

Dockerin-Cohesin Modeling process

We want to assess the binding affinity of cohesin and dockerin by calculating the equilibrium constant of the protein's association/dissociation from Gibbs free energy change during the binding process. Here we use the method defined by Wojciechowski et al. 2018. To be specific, two modes of binding are considered as the possible outcome for our cohesin and dockerin's association, their respective free energy changes are listed below as and .

In the two equations, the expression to be summed up consists of a Boltzmann factor; the difference between threshold energy and the free energy change of binding for a particular complex formed, which is dependent on the spatial orientation of the two proteins, characterized by the angle of rotation , and the distance between the two proteins measure on the vertical axis, Z; and a unit step function that takes the value of 1 when x>0 and the value of 0 when .

The first summation aims to find the total difference of energy when the complex formed takes different spatial orientation, the second summation factors in the separation between the two proteins, with a positive value meaning to shift the molecules closer together and a negative value of shifting them further apart, and the third one, containing index k, means to sum up the previously yielded sum for every suitable complex selected from the table shown below, finally we take the logarithm of the total sum and then times to get the effective free energy change.

The Coh-Doc complexes comprising the full-length Doc-table adopted from literature
The Coh-Doc complexes comprising the full-length Doc-table adopted from literature

In this table, and represent the cohesin derived from PDB:1OHZ and PDB:2CCL respectively, , represent the truncated dockerin derived from PDB:1OHZ and PDB:2CCL respectively, and , represent the dockerin in full-length, derived from PDB:1OHZ and PDB:2CCL. The asterisk denotes the dockerin with two point mutations S45A and T46A. (After sequence alignment we found that the cohesin and dockerin we used in experiments match the model PDB:1OHZ better, hence we shall consider only and , and as we only used wild type cohesin and dockerin we won't consider the mutant type in the calculations.)

Alignment of Cohesin 1, where the amino acid sequence sequences are 100% matched.
Alignment of Cohesin 1, where the amino acid sequence sequences are 100% matched.

Alignment of Dockerin 1, where about 50% amino acid sequences are matched.
Alignment of Dockerin 1, where about 50% amino acid sequences are matched.

To calculate the equilibrium constant of binding/dissociation, the general expression for Gibbs free energy is first used and then rearrange it to make K the subject:

Results and Discussions

This simple yet solid model confirms that cohesin and dockerin has a high binding affinity, this proves that our design of assembling the cellulosome complex is realistic and applicable.

As the minimum change in Gibbs Free energy in mode I is much higher in value than that in mode II, it is safe to assert that the binding between cohesin I and dockerin I mainly adopts mode I. Equilibrium constant calculated based on mode I:

After the modelling of cohesin-dockerin binding backed up our experimental design, we decided to attempt the fusion of RFP and dockerin I again by using different methods. More specifically, we changed the chemical used to induce RFP expression by E.coli from IPTG to Rhamnose, at the same time using a new version of RFP called eforRED. This time we are finally successful and the picture above shows the binding of RFP and dockerin I.
After the modelling of cohesin-dockerin binding backed up our experimental design, we decided to attempt the fusion of RFP and dockerin I again by using different methods. More specifically, we changed the chemical used to induce RFP expression by E.coli from IPTG to Rhamnose, at the same time using a new version of RFP called eforRED. This time we are finally successful and the picture above shows the binding of RFP and dockerin I.

3D model of cohesin-dockerin interaction generated by AlphaFold multimer

Dockerin I — Cohesin I
Dockerin I — Cohesin I

Dockerin I fused with eforRED
Dockerin I fused with eforRED

Model for Lytic polysaccharide monooxygenases

Background

As a cellulase booster, LPMO is able to open the three-dimensional network formed by polysaccharides like cellulose and chitin by hydrolysing the polymer into oligomers, [Cellobiose dehydrogenase Florian Csarman†, Lena Wohlschlager†, Roland Ludwig*] playing an important role in polysaccharide degradation. Unfortunately, our team has not yet expressed this enzyme, thus we resort to the model as an alternative stimulation.

In order to quantitatively evaluate the working efficiency of LPMO, we constructed a model to characterize its kinetic behaviour.

Mechanism of LPMO: A) Receiving an electron from Cellobiose dehydrogenase CDH or other electron donors like ascorbic acid, B) substrate binding is likely preceded by the reduction of the ground-state LPMO-Cu(II) to LPMO-Cu(I). C) Then, LPMOs conduct hydroxylation to cleave the glycosidic bonds, introducing chain breaks in the crystalline regions of densely packed cellulose fibrils and thereby providing new ends for processive hydrolases and promoting loosening of the cellulose structure. D) Then, LPMOs are oxidised by the hydrogen peroxide or oxygen back to their inactivate state LPMO-Cu(II), ready for the next binding. [Molecular mechanism of the chitinolytic peroxygenase reaction ]
Mechanism of LPMO: A) Receiving an electron from Cellobiose dehydrogenase CDH or other electron donors like ascorbic acid, B) substrate binding is likely preceded by the reduction of the ground-state LPMO-Cu(II) to LPMO-Cu(I). C) Then, LPMOs conduct hydroxylation to cleave the glycosidic bonds, introducing chain breaks in the crystalline regions of densely packed cellulose fibrils and thereby providing new ends for processive hydrolases and promoting loosening of the cellulose structure. D) Then, LPMOs are oxidised by the hydrogen peroxide or oxygen back to their inactivate state LPMO-Cu(II), ready for the next binding. [Molecular mechanism of the chitinolytic peroxygenase reaction ]

This model begins with the production of the two proteins (LPMO and CDH) involved. LPMO is our working enzyme and CDH is its booster which serves the function of transferring an electron to LPMO at the initial stage of the reactions.

Characterise LPMO and CDH by ordinary differential equations (ODE)

Transcription of the gene coding for LPMO:

Translation of LPMO's mRNA:

Transcription of the gene coding for CDH:

Translation of CDH's mRNA:

In these ODEs, and are the rate constant of transcription of the relevant DNA, , are the rate constants of translation of the relevant mRNAs, and and are the rate constant for mRNA and protein degradation, for simplicity's sake we assume that these constants are the same for LPMO and CDH.

Start off the reactions

The first reaction, dubbed “priming reduction”, involves the transfer of one electron from CDH to LPMO, this redox reaction can be shown by the following chemical equation, where II and I symbolize the copper ion bound to LPMO having an oxidation state of 2 and 1 respectively, and LPMO (I) is the activated form of the enzyme.

Describe this equation in ODE:

In this function, is the rate constant for reduction of LPMO.

After the enzyme is activated, it needs to associate with the cellulose polymer first. In order to describe this process, we adopt the Langmuir adsorption isotherm. The rationale behind this is: cellulose is a polymer that's insoluble in the water, so the binding of LPMO with it is actually an adsorption process in which the enzyme attaches to the surface of its substrate. In the equation shown below C is the number of available binding sites on cellulose, is the number of total binding sites on cellulose, and is the concentration of bound enzyme.

The ODE used to describe it can be written as:

The relationship applied here is . In the ODE, is the rate constant for enzyme adsorption and is the rate constant for enzyme desorption.

After the enzyme associates with cellulose the first step of our major reaction (hydroxylation of cellulose) can start:

This is a bisubstrate reaction and its kinetic mechanism follows the sequential binding model, so we construct the bisubstrate rate law as:

In this expression, is the Michaelis constant for hydrogen peroxide, is the Michaelis constant for cellulose and is the catalytic constant for the oxidative cleavage.

Following this reaction, the hydroxylated cellulose can go through an enzyme-independent cleavage to form the products, which are essentially two shorter chains of cellulose:

Results and discussions

We used the “deSolve” package in R to solve the ODEs listed above. And the results yielded are presented here using “ggplot” in R.

This graph shows the variation of the amount of LPMO and CDH over time. Overall their growth follows a smooth sigmoidal curve, with the increase in CDH always above that of LPMO, possibly because the transcription rate of CDH is significantly higher than that of LPMO. Furthermore, the graph shows a tendency to level off as time progresses, this can be explained by the fact that active form of LPMO and CDH are consumed respectively by adsorbing onto the cellulose (LPMO), and being reduced in the priming reduction step (CDH). Hence it can be predicted that the amount of LPMO and CDH will reach a plateau eventually.
This graph shows the variation of the amount of LPMO and CDH over time. Overall their growth follows a smooth sigmoidal curve, with the increase in CDH always above that of LPMO, possibly because the transcription rate of CDH is significantly higher than that of LPMO. Furthermore, the graph shows a tendency to level off as time progresses, this can be explained by the fact that active form of LPMO and CDH are consumed respectively by adsorbing onto the cellulose (LPMO), and being reduced in the priming reduction step (CDH). Hence it can be predicted that the amount of LPMO and CDH will reach a plateau eventually.

For the production of hydroxylated cellulose: The graph generated shows that initially the formation of product is negligible, this can be rationalized by considering the sequence of events taking place: In the beginning, LPMO molecules are being prepared for reaction by being reduced, then they need to adsorb onto the cellulose substrate. So it is understandable that the initial rate of product formation is negligible—the amount of enzyme present to catalyzed the hydroxylation reaction is scarce. However, after the initial period, rate of product formation gradually approaches a steady value (the graph starts to be linear). This is a welcomed result as it demonstrates that the hydroxylation reaction is proceeding at a constant rate, there is no evidence for feedback inhibition or enzyme denaturation — by-products do not significantly alter the thermodynamic property of the reaction system, nor do them impact the pH of the mixture to a visible extent.
For the production of hydroxylated cellulose: The graph generated shows that initially the formation of product is negligible, this can be rationalized by considering the sequence of events taking place: In the beginning, LPMO molecules are being prepared for reaction by being reduced, then they need to adsorb onto the cellulose substrate. So it is understandable that the initial rate of product formation is negligible—the amount of enzyme present to catalyzed the hydroxylation reaction is scarce. However, after the initial period, rate of product formation gradually approaches a steady value (the graph starts to be linear). This is a welcomed result as it demonstrates that the hydroxylation reaction is proceeding at a constant rate, there is no evidence for feedback inhibition or enzyme denaturation — by-products do not significantly alter the thermodynamic property of the reaction system, nor do them impact the pH of the mixture to a visible extent.

Flaws of model design and possible improvements

First of all, although our model successfully elucidated the kinetic behaviour of LPMO and showed it can oxidatively cleave its cellulose polymer substrate at an appreciable rate, it doesn't account for further reactions that will take place catalyzed by the other cellulases, i.e. endoglucanase, exoglucanase and β-glucosidase. In the future, it will certainly be enlightening if a model that takes the action of these three cellulases into consideration can be integrated into ours.

Secondly, our model does show that the initial rate of product formation is very low. To accelerate this process and make the degradation of cellulose by cellulosome enzymes more time-efficient, it is necessary to increase LPMO and CDH's transcription rates by using stronger promoters. Furthermore, the rate constant for the reduction reaction is choosen to be quite low when we are plugging in the parameters, this is not ideal since a low rate of reduction means the priming reaction alone would take some time to generate enough activated LPMO. To change this may not be easy, as increasing the rate constant of reduction possibly requires the affnity between LPMO and CDH to be higher. It is probable that through engineering the two proteins' interacting domains that facilitated their association, one can increase and make the priming reduction step more suitable if out project is to be applied on an industrial scale.

References

  1. Kuusk, Silja, et al. “Kinetics of H2O2-driven Degradation of Chitin by a Bacterial Lytic Polysaccharide Monooxygenase.” Journal of Biological Chemistry, vol. 293, no. 2, Elsevier BV, Jan. 2018, pp. 523-31. DOI: 10.1074/jbc.m117.817593.
  2. Anuganti, Murali, et al. “Kinetic Study on Enzymatic Hydrolysis of Cellulose in an Open, Inhibition-Free System.” Langmuir, vol. 37, no. 17, American Chemical Society (ACS), Apr. 2021, pp. 5180-92. DOI: 10.1021/acs.langmuir.1c00115.
  3. Bissaro, Bastien, et al. “Molecular Mechanism of the Chitinolytic Peroxygenase Reaction.” Proceedings of the National Academy of Sciences, vol. 117, no. 3, Proceedings of the National Academy of Sciences, Jan. 2020, pp. 1504-13. DOI: 10.1073/pnas.1904889117
  4. Koš, Martin, and David Tollervey. “Yeast Pre-rRNA Processing and Modification Occur Cotranscriptionally.” Molecular Cell, vol. 37, no. 6, Elsevier BV, Mar. 2010, pp. 809-20. DOI: 10.1016/j.molcel.2010.02.024
  5. Gao, Dahai, et al. “Increased Enzyme Binding to Substrate Is Not Necessary for More Efficient Cellulose Hydrolysis.” Proceedings of the National Academy of Sciences, vol. 110, no. 27, Proceedings of the National Academy of Sciences, June 2013, pp. 10922-27. DOI: 10.1073/pnas.1213426110.
  6. Rajeshkannan, et al. “GAL Regulon in the Yeast S. Cerevisiae Is Highly Evolvable via Acquisition in the Coding Regions of the Regulatory Elements of the Network.” Frontiers in Molecular Biosciences, vol. 9, Frontiers Media SA, Mar. 2022, DOI: 10.3389/fmolb.2022.801011.
  7. Wojciechowski, Michał, et al. “Dual Binding in Cohesin-dockerin Complexes: The Energy Landscape and the Role of Short, Terminal Segments of the Dockerin Module.” Scientific Reports, vol. 8, no. 1, Springer Science and Business Media LLC, Mar. 2018, DOI: 10.1038/s41598-018-23380-9.