Alphafold

Introduction

Our initial idea was centred around the use of RNAIII inhibitory peptide (RIP) to competitively inhibit RNAIII activating peptide (RAP) binding to the target of RAP (TRAP) ^[1]. According to various sources we found, this would downregulate the agr system, linked to autoinducing peptide (AIP) based quorum sensing (QS). Leading to reduced expression of RNAIII which studies had linked to reduced biofilm formation ^[2].

Prior to forming our model, we hit some blocks when searching for the DNA sequence of RAP.

We were unable to find any sequence for 'RAP' or 'RNAIII inhibitory peptide' across multiple gene and protein databases, including both the gene and protein database in NCBI and the protein database in UniProt.
The sequence we managed to find was 'Large ribosomal subunit protein uL2' which had 100% amino acid identity with the RAP sequence given in a paper by Korem et al. who generated a recombinant form of the protein to characterise RAP for the first time ^[3].
When we searched 'Large ribosomal subunit protein uL2' in both NCBI and UniProt protein databases again we found a sequence, however RAP was not listed as an alternative name.
This began to raise some flags, but we had begun some modelling alongside, which helped us test the likelihood that these proteins would actually interact.

Model

An image of RAP aligned to 50S ribosomal protein L2 using the matchmaker tool in the visualisation program ChimeraX

Figure 1: (Blue) RAP best model from Alphafold 2 aligned to (orange) 50S ribosomal protein L2 using matchmaker tool in ChimeraX. Both amino acid sequences were sequenced from Staphylococcus aureus.
The Structure of RAP

Using the matchmaker command in ChimeraX the protein structures for RAP and 50S ribosomal protein uL2 have been superimposed using a pairwise sequence alignment and then by fitting the aligned residue pairs.
The RMSD (root-mean-square deviation) between the two sets of atoms is 0.243Å. For reference, a successful RMSD score is less than or equal to 1.5Å ^[4]. Providing confirmation that the 50S ribosomal binding protein uL2 was indeed the protein sequence that had been previously labelled as RAP.
The per residue confidence scores (pLDDT) for both proteins were mostly above 90, indicating a very high confidence score.

We then modelled the binding of RAP to TRAP using the Alphafold tool integrated into ChimeraX (predict function).

The Binding of RAP and TRAP

According to the mechanism proposed by Balaban, et al. ^[5] who claimed to discover RIP in 1998 ^[6] TRAP is the receptor for RAP. Therefore, we wanted to model the protein-protein interactions between RAP and TRAP to see how they would bind. Our initial thinking behind this was that we could then see how RIP would bind, in order to better understand the mechanism of competitive inhibition.
As we started this process the first red flag to appear was that the sequence for RAP was much longer than that of the receptor, TRAP.
- RAP = 277 aa
- TRAP = 167 aa
Figure 2: Best model prediction for interaction between RAP (PAE domains in pink, green, yellow) and TRAP (PAE domain blue)
This could have been explained if TRAP was a multimer, but on RSCB PDB the global stoichiometry was found to be monomeric in both solution and the crystalline form.
PAE (predicted aligned error) is a useful tool for assessing the quality of inter-domain predictions. Where pLDDT is very useful for assessing the quality of individual domain structure predictions (see structure of RAP discussion), PAE has been used here ^[7].

Figure 3: PAE matrix for the interaction between domains in the RAP/TRAP complex, generated using PAE viewer ^[8].

The PAE is generally high (10-30 Å) - see Figure 3. This indicates that the relative position and orientation of RAP and TRAP is uncertain, meaning it should not be interpreted ^[7].
These results flagged to us that it is unlikely that RAP and TRAP do bind, suggesting that these systems may not function as described in the literature we had found.
Initially we thought this could mean TRAP was not the receptor, and that there was an uncharacterised sensor kinase - with the proposed mechanism being RAP binding to the SK leading to the phosphorylation of the HK domain in TRAP.

Returning to the Literature

The output from Alphafold2 and ChimeraX, paired with the struggles we faced when searching for the sequences initially, lead us to become sceptical of this mechanism. Upon returning to the literature, we found various contradictory papers. This discourse was summarised in a 2007 article ^[9]. To summarise, Balaban and Novick published the first paper on RIP in 1995. Subsequent experiments carried out in the same year found that the agr-stimulating activity which had been attributed to RAP could actually be due to the action of AIPs. This was disputed by Balaban's group in a 1998 paper. In the following years more papers were published on the subject, however the nail in the head came with a paper lead by Novick ^[10], who was an author on the first paper. This paper determined that Balaban's results could be due to AIP contamination rather than the action of RIP. Following this, Balaban published a sub-par response. This was followed by the article in which we found the 'Large ribosomal subunit protein uL2' DNA sequence ^[3], bringing us in a full circle. To summarise all the findings: inactivation of traP (the gene for TRAP) has no impact on S. aureus virulence (specifically the haemolytic activity), and disruption of TRAP had no effect on S. aureus surface protein profile ^[11]. Therefore, disproving Balaban's work. A few years later a paper was released reviewing these findings, and interrogating whether RIP could be effective ^[12]. The results showed that TRAP may function to reduce oxidative stress, and that RIP could be effective - but only with an amide group on the C-terminus. This is not possible to encode and replicate in bacteria. Therefore, we returned to idea generation - using the skills and knowledge we had gained to find a way to inhibit biofilm formation.

References

Gov Y, Bitler A, Dell'Acqua G, Torres JV, Balaban N. RNAIII inhibiting peptide (RIP), a global inhibitor of Staphylococcus aureus pathogenesis: structure and function analysis. Peptides. 2001;22(10):1609-20.
De Oliveira A, Pinheiro-Hubinger L, Pereira VC, Riboli DFM, Martins KB, Romero LC, Cunha MDLRDSD. Staphylococcal Biofilm on the Surface of Catheters: Electron Microscopy Evaluation of the Inhibition of Biofilm Growth by RNAIII Inhibiting Peptide. Antibiotics. 2021;10(7):879.
Korem M, Sheoran AS, Gov Y, Tzipori S, Borovok I, Balaban N. Characterization of RAP, a quorum sensing activator of Staphylococcus aureus. FEMS Microbiology Letters. 2003;223(2):167-75.
Hevener KE, Zhao W, Ball DM, Babaoglu K, Qi J, White SW, Lee RE. Validation of Molecular Docking Programs for Virtual Screening against Dihydropteroate Synthase. Journal of Chemical Information and Modeling. 2009;49(2):444-60.
Balaban N, Goldkorn T, Gov Y, Hirshberg M, Koyfman N, Matthews HR, et al. Regulation of Staphylococcus aureus Pathogenesis via Target of RNAIII-activating Protein (TRAP). Journal of Biological Chemistry. 2001;276(4):2658-67.
Balaban N, Goldkorn T, Nhan RT, Dang LB, Scott S, Ridgley RM, et al. Autoinducer of virulence as a target for vaccine and therapy against Staphylococcus aureus. Science. 1998;280(5362):438-40.
AlphaFold Protein Structure Database. Frequently asked questions: EMBL European Bioinformatics Institute; [Internet]. Available from: https://alphafold.ebi.ac.uk/faq.
Elfmann C, Stülke J. PAE viewer: a webserver for the interactive visualization of the predicted aligned error for multimer structure predictions and crosslinks. Nucleic Acids Research. 2023;51(W1):W404-W10.
Ahmad S, Rahman H, Qasim M, Nawab J, Alzahrani KJ, Alsharif KF, Alzahrani FM. Staphylococcus epidermidis Pathogenesis: Interplay of icaADBC Operon and MSCRAMMs in Biofilm Formation of Isolates from Pediatric Bacteremia in Peshawar, Pakistan. Medicina. 2022;58(11):1510.
Novick RP, Ross HF, Figueiredo AMS, Abramochkin G, Muir T. Activation and Inhibition of the Staphylococcal AGR System. Science. 2000;287(5452):391-.
Shaw LN, Jonsson I-M, Singh VK, Tarkowski A, Stewart GC. Inactivation of traP Has No Effect on the Agr Quorum-Sensing System or Virulence of Staphylococcus aureus. Infection and Immunity. 2007;75(9):4519-27.
Kiran MD, Balaban N. TRAP Plays a Role in Stress Response in Staphylococcus Aureus. The International Journal of Artificial Organs. 2009;32(9):592-9.

RIP-RAP-TRAP

Introduction

In order to further understand our project, we built a model of the expression of RNAIII due to an increase of RNA inhibitory protein, RIP. To do this, we built a model of differential equations based on the following mechanism^[1] through the 2 transcripts of the agr operon, RNAII and RNAIII.

Production of autoinducing peptides (AIPs) from AgrB and AgrD was simplified by modelling RNAII transcription to AIP directly. AgrA phosphorylation following AgrC phosphorylation by AIP was also simplified by assuming that AIPs directly phosphorylated AgrA. Production of autoinducing peptides (AIPs) from AgrB and AgrD was simplified by modelling RNAII transcription to AIP directly. AgrA phosphorylation following AgrC phosphorylation by AIP was also simplified by assuming that AIPs directly phosphorylated AgrA.

We then built a model involving 2 separate components, the RIP-RAP-TRAP mechanism, and the expression of RNAII and RNAIII through the autoinduction by AIPs.

Model

RIP RAP TRAP mechanism

To develop the first part of our model, we assumed that all reactions were one -to-one reactions. We also assumed that RIP binding to TRAP and RAP binding to TRAP were both reversible reactions. From these two assumptions we get the series of ordinary differential equations:

$$\frac{\mathrm{d} [RIP]}{\mathrm{d} t} = -k_{1}([RIP]\cdot[TRAP]) +t_{1}\cdot[TRAPR]$$

$$\frac{\mathrm{d} [RAP]}{\mathrm{d} t} = -k_{2}([RAP]\cdot[TRAP]) +t_{2}\cdot[TRAPP]$$

$$\frac{\mathrm{d} [TRAPR]}{\mathrm{d} t} = k_{1}([RIP]\cdot[TRAP]) -t_{1}\cdot[TRAPR]$$

$$\frac{\mathrm{d} [TRAPP]}{\mathrm{d} t} = k_{2}([RAP]\cdot[TRAP]) -t_{2}\cdot[TRAPP] $$

$$\frac{\mathrm{d} [TRAP]}{\mathrm{d} t}=-k_2 ([RAP]\cdot[TRAP])+t_2\cdot[TRAPP]-k_1 ([RIP]\cdot[TRAP])+t_1\cdot[TRAPR]$$

With this set of differential equations, we can then move onto the next part of our model with phosphorylated TRAP upregulating the agr operon, and subsequently RNA III.

Agr operon

Phosphorylated TRAP upregulates the agr system through a poorly characterised mechanism. The P2 promoter is activated, leading to RNAII expression, resulting in the downstream production of AIP and AgrA phosphorylation. TRAP phosphorylation can thus be simplified as triggering the production of AIPs and AgrA phosphorylation. This can be modelled as a chemical reaction relating AIPs and AgrA. AIPs phosphorylate AgrA which then activates transcription from both the P2 and P3 promoters for RNAII and RNAIII respectively. It is assumed that once AgrA is phosphorylated and activates the promoters, it becomes dephosphorylated. AIPs also trigger the dephosphorylation of TRAP^[2]. Both these reactions were modelled with the following differential equations:

$$\frac{\mathrm{d} [AIP]}{\mathrm{d} t}=-k_6\ ([AIP]\cdot[AgrA])+k_3 ([TRAPP]\cdot[p2])+k_9 ([AgrAP]\cdot[p3])$$

$$\frac{\mathrm{d} [AgrA]}{\mathrm{d} t}=-k_6 ([AIP]\cdot[AgrA])+k_3 ([TRAPP]\cdot[p2])+k_9 ([AgrAP]\cdot[p3])+k_8 ([AgrAP]\cdot[p3])$$

$$\frac{\mathrm{d} [AgrAP]}{\mathrm{d} t}=k_6 ([AIP]\cdot[AgrA])-k_9 ([AgrAP]\cdot[p3])-k_8 ([AgrAP]\cdot[p3]) $$

$$\frac{\mathrm{d} RNAIII}{\mathrm{d} t}=k_8\ ([AgrAP]\cdot[p3]) $$

Combining these equations with those for the RIP-RAP-TRAP mechanism previously defined, we get:

$$\frac{\mathrm{d} [TRAPP]}{\mathrm{d} t}=k_2 ([RAP]\cdot[TRAP])-t_2\cdot[TRAPP]-k_7 ([AIP]\cdot[TRAPP])$$

$$\frac{\mathrm{d} [TRAP]}{\mathrm{d} t}=-k_2 ([RAP]\cdot[TRAP])+t_2\cdot[TRAPP]-k_1 ([RIP]\cdot[TRAP])+t_1\cdot[TRAPR]+k_7 ([AIP]\cdot[TRAPp])$$

Degradation and production rates

Since the molecules degrade over time, degradation rates for each of the molecules needs to be considered. Since TRAP, RIP, and AIP are also formed outside the modelled mechanism, these considerations need to be incorporated. This results in the following equations:

$$\frac{\mathrm{d} [RIP]}{\mathrm{d} t} = -k_{1}([RIP]\cdot[TRAP]) +t_{1}\cdot[TRAPR] + secretion \:rate$$

$$\frac{\mathrm{d} [RAP]}{\mathrm{d} t} = -k_{2}([RAP]\cdot[TRAP]) +t_{2}\cdot[TRAPP]$$

$$\frac{\mathrm{d} [TRAP]}{\mathrm{d} t}=-k_2 ([RAP]\cdot[TRAP])+t_2\cdot[TRAPP]-k_1 ([RIP]\cdot[TRAP])+t_1\cdot[TRAPR] + TRAP\:production\:rate$$

$$\frac{\mathrm{d} [TRAPP]}{\mathrm{d} t}=k_2 ([RAP]\cdot[TRAP])-t_2\cdot[TRAPP]-k_7 ([AIP]\cdot[TRAPP])$$

$$\frac{\mathrm{d} [TRAPR]}{\mathrm{d} t} = k_{1}([RIP]\cdot[TRAP]) -t_{1}\cdot[TRAPR]$$

$$\frac{\mathrm{d} [AgrA]}{\mathrm{d} t}=-k_6 ([AIP]\cdot[AgrA])+k_3 ([TRAPP]\cdot[p2])+k_9 ([AgrAP]\cdot[p3])+k_8 ([AgrAp]\cdot[p3])$$

$$\frac{\mathrm{d} [AgrAP]}{\mathrm{d} t}=k_6 ([AIP]\cdot[AgrA])-k_9 ([AgrAP]\cdot[p3])-k_8 ([AgrAP]\cdot[p3])$$

$$\frac{\mathrm{d} [AIP]}{\mathrm{d} t}=-k_6\ ([AIP]\cdot[AgrA])+k_3 ([TRAPP]\cdot[p2])+k_9 ([AgrAP]\cdot[p3]) + AIP \:production \:rate$$

$$\frac{\mathrm{d} RNAIII}{\mathrm{d} t}=k_8\ ([AgrAP]\cdot[p3])$$

Constants

Constant	Reaction constant for
K₁	RIP binding to TRAP
K₂	RAP binding to TRAP
K₃	TRAPP causing the production of AIPS
K₆	AIP phosphorylating AgrA
K₇	AIP dephosphorylating TRAPP
K₈	Expression of RNAIII by AgrAP
K₉	Expression of RNAII (production of AIPs) by AgrAP
Constant	Utility
t₁	Rate of TRAPR to TRAP + RIP
t₂	Rate of TRAPP to TRAP + RAP
[p2]	Concentration of p2
[p3]	Concentration of p3
Constant	Utility
Secretion Rate	Secretion rate of RIP by the bacteria
TRAP Production rate	Production of TRAP not by the mechanism
AIP Production rate	Production of AIP not by the mechanism
Constant	Degradation rate for
d₁	RIP
d₂	RAP
d₃	TRAP
d₄	TRAPP
d₅	TRAPR
d₆	AgrA
d₇	AgrAP
d₈	RNAIII

Conclusion

graph showing an exponential increase of RNAIII activity with time, There are two lines, One with no scretion of rip, one with secretion of rip. The one with secretion of rip is shallower

We ran this model on python using scipy.integrate.odeint

We graphed this over time and compared RNAIII expression with RIP secretion and with no RIP secretion to determine the shape of the graph before finding any of the constants.

From the shape of the graph, an increase in RIP secretion leads to a decrease in RNAIII expression. This shape stays constant even with different constants.

Concerns

However, our protein structure model lead to doubts regarding whether RIP/RAP bounded to TRAP, which was the main assumption in our RIP-RAP-TRAP model above. Additionally, Vuong et al.^[3] noted agr deletion mutants increased biofilm formation, which lead us to have some doubts about the RIP-RAP-TRAP system and its antibiofilm effect.

References

De Oliveira A, Pinheiro-Hubinger L, Pereira VC, Riboli DFM, Martins KB, Romero LC, et al. Staphylococcal Biofilm on the Surface of Catheters: Electron Microscopy Evaluation of the Inhibition of Biofilm Growth by RNAIII Inhibiting Peptide. Antibiotics. 2021;10(7):879. DOI: https://doi.org/10.3390/antibiotics10070879
Kumar, Nishant & Gupta, Hansita & Dhasmana, Neha & Singh, Yogendra. (2018). Combating Staphylococcal Infections Through Quorum Sensing Inhibitors. DOI: 10.1007/978-981-10-9026-4_15
Cuong Vuong and others, Quorum-Sensing Control of Biofilm Factors in Staphylococcus epidermidis, The Journal of Infectious Diseases, Volume 188, Issue 5, 1 September 2003, Pages 706–718. DOI: https://doi.org/10.1086/377239

DPD

Introduction

One of the aims of our project is to increase expression of S-ribosylhomocysteinase enzyme (LuxS) in L. Plantarum, to increase autoinducer 2 (AI-2) molecules and decrease biofilm formation downstream, as previously explained in our Project Description.

LuxS catalyses the reaction of S-ribosyl-L-homocysteine (Srh) to Homocysteine (Hcy) and 4,5-dihydroxy-2,3-pentanedione (DPD). Hcy gets recycled back to Srh and DPD acts as the precursor for AI-2 through various intermediate stages.

Whether the amount of Srh needs to be replenished to achieve an increase in DPD production, and thus AI-2, is a question we hope to answer using our model.

Mechanism:

Model

Our model looks at the relationship between an increase in LuxS and DPD production. To build this model, the mechanism of Srh and DPD production are needed. The model assumes an increase in DPD will increase AI-2 and excludes other ways of DPD production not from Srh.

Additionally, the production of Srh from Hcy is considered direct and the intermediate stages ignored. Using the four-molecule system (Srh, LuxS, DPD, Hcy) a set of differential equations were built.

Differential Equations

To develop the set of equations, the main reaction, DPD production from Srh catalysed by LuxS was modelled using the Henri Michaelis Menten equation^[1]:

$$\frac{V_{max}\cdot[S]}{k_m+[S]}$$

V_max is defined as: V_max = k₂⋅[E_t], where [E_t] is the concentration of enzyme both attached to the substrate and not attached to the substrate^[1]. The model was based on differing amounts of LuxS ([E_t]) and thus V_max was separated in the model, resulting in the reaction velocity equation:

$$\frac{k_2\cdot[E_2]\cdot[S]}{k_m+[S]}$$

The production of Srh from Hcy was assumed as a first order reaction due to it solely being based on the concentration of Hcy. Thus, simple equation used was:

$$v\ =P_1\cdot[Hcy]$$

There are two molecules in the model which are produced from outside the mechanism: Hcy and LuxS. It was assumed the production of these two molecules from outside the system did not rely on any components produced in the system.

Degradation constants for all molecules except DPD were incorporated, as we wanted to find the total DPD produced, and assumed its concentration did not affect any other part of the mechanism. This leaves the following set of ordinary differential equations:

$$\frac{\mathrm{d} DPD}{\mathrm{d} t}=\frac{k_2\cdot[E_t]\cdot[S]}{k_m+[S]}\cdot V$$

$$\frac{\mathrm{d} [S]}{\mathrm{d} t}=(P_1\cdot[Hcy]-\frac{k_2\cdot[E_t]\cdot[S]}{k_m+[S]})\cdot V - P_2\cdot[S]$$

$$\frac{\mathrm{d} [Hcy]}{\mathrm{d} t}=(\frac{k_2\cdot[E_t]\cdot[S]}{k_m+[S]}-P_1\cdot[Hcy])\cdot V +[Hcy]\ production\ rate-P_3\cdot [Hcy]$$

$$\frac{\mathrm{d} [E_t]}{\mathrm{d} t}=E_t\ production\ rate-P_4\cdot [E_t]$$

Constants

	Symbol	Value/Equation	Unit	Reference/Assumption
Rate Constants	k_m	0.38 * 6.02e¹⁷	Molecules/L	Part: BBa_K091109 Registry of standard biological parts
	k₂	5.41e^-4	s^-1	Part: BBa_K091109 Registry of standard biological parts
	P₁	0.00017982625	s^-1	Determined from iterations of model to give a constant value of S for the wild type model
Production Rates	Hcy production rate	See equation below	Molecules/s	Matches the production of Hcy of the wild type to produce a constant concentration Hcy of 615 molecules
	E_t production rate	2.05	Molecules/s	^[5]^[6] Although 5 is for in Vibrio fischeri assumed similar in L plantarum
Initial Values	E_t0	205	Molecules	E_t production rate and P₄ assumed to reach an equilibirum immediately. Assumed 'normal' value of E_t
	S₀	600	Molecules	Assumed to be similar to Hcy₀
	DPD₀	0	Molecules	For ease measuring DPD production
	Hcy₀	615	Molecules	[2]
Degradation rates	P₂	7.467e^-12	N/A	[3]
	P₃	7.467e^-12	N/A	[3]
	P₄	0.01	N/A	Estimated to provide a constant amount of E_t in the wildtype
Miscellaneous	V	2.5e^-18	L	[3]
	Time	7.6e⁶	S	[4]

$$Hcy\:production\:rate=(-\frac{(k_2\cdot [E_t ]\cdot \frac{600}{V})}{k_m+\frac{600}{V}} +P_1\cdot \frac{615}{V})\cdot V+P_3\cdot 615$$

This model was run on python using scipy.integrate.odeint

Conclusion

Graph showing that the production of DPD increases with an increase in LuxS and increases even more with both increasing LuxS and Srh

The results show that an increase in the amount LuxS enzyme would result in an increase in DPD production. This is most likely due to the substrate (Srh) being formed from one of the by-products (Hcy) of the reaction catalysed by LuxS. The results also show that an increase in amount of Srh increases the production of DPD, but this is not required to see an increase in DPD production.

Based on the model, increasing the production of LuxS would not require Srh to be supplemented for DPD production to increase. This informed us that we would not need to replenish our L.plantarum expressing LuxS with Srh to increase the production of DPD, and thus AI-2.

References

Punekar NS. Henri-Michaelis-Menten Equation. ENZYMES: Catalysis, Kinetics and Mechanisms [Internet]. 2018 Nov 12 [cited 2023 Jul 20];155–76. Available from: https://link.springer.com/chapter/10.1007/978-981-13-0785-0_15
Park JO, Rubin SA, Xu YF, Amador-Noguez D, Fan J, Shlomi T, et al. Metabolite concentrations, fluxes and free energies imply efficient enzyme usage. Nature Chemical Biology [Internet]. 2016 May 2 [cited 2023 Jul 20];12(7):482–9. Available from: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4912430/
Lynch M, Marinov GK. The bioenergetic costs of a gene. Proceedings of the National Academy of Sciences [Internet]. 2015 Nov 2 [cited 2023 Jul 20];201514974. Available from: https://www.pnas.org/doi/full/10.1073/pnas.1514974112
Living with a urinary catheter - NHS [Internet]. nhs.uk. 2017 [cited 2023 Jul 20]. Available from: https://www.nhs.uk/conditions/urinary-catheters/living-with/#:~:text=The%20catheter%20itself%20will%20need
Lupp C, Ruby EG. Vibrio fischeri LuxS and AinS: Comparative Study of Two Signal Synthases. Journal of Bacteriology [Internet]. 2004 Jun 15 [cited 2023 Jul 20];186(12):3873–81. Available from: https://journals.asm.org/doi/pdf/10.1128/jb.186.12.3873-3881.2004
Garcia Hernan G, Lee H, Boedicker James Q, Phillips R. Comparison and Calibration of Different Reporters for Quantitative Analysis of Gene Expression. Biophysical Journal [Internet]. 2011 Aug [cited 2023 Jul 20];101(3):535–44. Available from: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3145315/

2D DIFFUSION

Introduction

As our initial idea was to attempt to form a biofilm-like bacterial coating on the surface of a catheter, we needed to check the viability of that as a method of delivering molecules to pathogenic bacteria in the environment

In order to gain an initial idea of the timings and pressures involved in this we decided to look through a lens of diffusion. If our molecules wouldn't be able to diffuse out of the bacterial coating and into the envirnonment, or if barely anything diffused into pathogenic bacteria then we would have to look at other delivery methods.

Model

To investigate diffusion in a very general way we decided to use Fick's diffusion model.

A diagram with 5 differently coloured boxes of varying sizes, they are arranged such that they divide a rectangle and are each touching at most 2 other boxes.
in order left to right they are labelled: catheter surface,
Bacteria in coating, Matrix around bacterial coating, Environment, and Bacteria attatched to surface of coating
the box labelled bacterua attached to surface coating is placed inside the box labelled environment and touching
the box labelled matrix around bacterial coating. There are arrows between any touching boxes. the leftmost upper corner
of bacteria in coating is labelled A, the leftmost upper corner of matrix around bacterial coating is labeled M1 and the upper right
corner is labelled M2. The barrier between the environment and bacteria attatched to surface of coating is labelled E on the side of the environment
and B on the side of the bacteria

For this system the assumptions made were:

That the matrix accounted for approximately 80-90% of the coating volume but stretched over the entire biofilm width from the catheter surface to the environment.
That only passive diffusion occurs following Fick's laws of diffusion based on an ideal liquid
That the environment consisted of water.
That the system was closed and while molecules could diffuse into the environment, they would not disappear from the system entirely.
That molecules were evenly dispersed throughout bacteria of each type, i.e. all bacteria in the coating had the same concentration, and all bacteria external to the coating had a different single concentration
That no other processes would affect the movement of molecules

In order to model this system the diffusion based equations were:

$$dm_{AM}=\frac{-D_A\cdot S_A \cdot (c_{M1} - c_A)}{x_A}$$

$$dm_{MM} = \frac{-D_M \cdot S_M \cdot (c_{M2}-c_{M1})}{0.5x_{BF}}$$

$$dm_{MB} = \frac{-D_B \cdot S_{MB} \cdot (c_{B}-c_{M2})}{0.5x_{BF}}$$

$$dm_{ME}=\frac{-D_E \cdot (S_{ME}-S_{MB}) \cdot (c_{E}-c_{M2})}{x_{MEB}} $$

$$dm_{EB}= \frac{-D_B \cdot S_{EB} \cdot (c_{B}-c_{E})}{x_{B}} $$

And the system of differential equations was

$$dc_A = \frac{1}{V_A} \cdot (production rate - dm_{AM})$$

$$dc_{M1} = \frac{1}{0.5V_M} \cdot (dm_{AM} -dm{MM})$$

$$dc_{M2} = \frac{1}{0.5V_M} \cdot (dm_{MM} - dm_{MB} - dm_{ME})$$

$$dc_E = \frac{1}{V_E} \cdot (dm_{ME} - dm_{EB})$$

$$dc_B = \frac{1}{V_B} \cdot (dm_{MB} + dm_{EB} - degradation constant)$$

Symbol	Meaning	Symbol	Meaning
dm_AM	rate of mass (of small molecule e.g. ai-2) moving from bacterial coating (A) to matrix (M1)	dm_MM	rate of mass (of small molecule e.g. ai-2) moving from M1 to M2
dm_MB	rate of mass (of small molecule e.g. ai-2) moving from bacterial coating (A) to matrix (M1)	dm_ME	rate of mass (of small molecule e.g. ai-2) moving from matrix 2 (M2) to environment (E)
dm_EB	rate of mass moving from environment (E) to external bacteria (B)	D_x	Diffusion constant in x
S_x	Surface area of x	c_x
S_MB	Surface area of matrix (M2) touching only external bacteria (B)	S_ME	Surface area of matrix (M2) touching only environment (E)
dc_x	rate of change of concentration in x	V_x	Volume of x

Conclusion

When testing this model, we found that:

Changing the diffusion coefficient in the water/matrix ratio had a relatively small impact on the time taken to reach equilibrium
The permeability constants of the bacterial membranes had a significant impact on the time taken to diffuse, with a larger permeability coefficient leading to less time taken diffuse. However they had litte to no effect on the final concentrations reached when the system was at equilibrium.
This equilibrium was generally reached in a time of around 2-8 minutes.
The environmental volume had the greatest impact on the final concentrations that reached equilibrium.

Because of this we wanted to more accurately determine the permeability coefficiant as this seemed to have the greatest potentialy impact on the viability of our idea.

3D DIFFUSION

Introduction

The main weakness of the fick's diffusion based ODE model seemed to lay in the assumption of passive diffusion and the variability of the model based on membrane permeability.

Due to the variability of diffusive behaviour depending on molecular size, structure, and polarity, the easiest way to obtain more accuracy in this was via simulation-based molecular modelling.

Model

To do this we used GROMACS^[1]^[2] with the CHARMM-27 forcefield, first to view diffusion of ai-2 in water to get an aqueous diffusion constant, then to investigate the possibility of diffusion through a membrane.

We used the non-ionic form of ai2 as some research indicated it to be the most common and it would likely obtain the more accurate modelling results with the computational power available

Figure 1: The molecular diagram of ai-2.

In the aqueous diffusion model we obtained a diffusion constant of 0.33 +- 0.02e-5, which aligns with our expectation from literature values for similar molecules ^[3]
It should however be noted that GROMACS systems are periodic - if an atom reaches the boundary of the system it will appear on the other side of it. This can cause issues in modelling diffusion as enlarging the system's box means more molecules must be generated which increases the computational load (which can already take over 2 day to run one simulation), however leaving the box too small can lead to diffusion effects being unclear due to molecules moving into an area in reverse. This can be seen below in the result from aqueous diffusion.

Figure 2: A gif showing the random motion of ai-2 molecules. The molecules can be seen appearing at the top of the box due to periodicity, as well as diffusing gradually upwards.

While this is less of an issue in the ai-2 and water case, for the membrane this meant that just the GROMACS simulation alone could not be used to visually detect if diffusion was occurring through the membrane, as can be seen below.

In order to make a membrane model we used the Charmm standard membrane for inner membranes of gram negative bacteria ^[4] ^[5] and then used SwissParam to get molecule files in the correct format for each lipid. ^[6]

Figure 3: A gif showing a membrane (in the middle) and ai-2 starting in a block above the membrane and then diffusing outwards, after a few frames ai2- molecules can be seen appearing at the bottom of the image and these diffuse upwards until it is hard to tell whether they came through the membrane or not.

In order to remove any atoms that reached the other side of the membrane via this means a python based code was used to remove these atoms, while a full visualisation could not be created for this reduced set of atoms, the data showing distance travelled in terms of one axis can be plotted

a graph of atom trajectories showing lines which gradually move upwards. There is a pink rectangle at 10 to 15 nm and after 2000ps the atom trajectories start to overlap with it. one atom goes all the way through the membrane.

Figure 4: A graph of atom trajectories in the direction of diffusion towards and through the membrane.

This reduced dataset showed molecules being able to travel approximately 2.5nm into the membrane in a time of 15ns

From this we can use the Stokes-Einstein diffusion equation $$t_d = \frac{d^2}{2D} $$ (Where t_d is the average time taken for a molecule to diffuse over distance d, and D is the diffusion constant) to obtain a value for D of 2.08e-6 cm/s.

Conclusion

Returning to our previous Fick's Law ODE model and using this value and the membrane width used for the simulation to give the permeability coefficient we see that 80% of the ai-2 would diffuse out of the bacteria in the coating within around 3 seconds (with only very little correlation found between the values other constants and time to reach equilibrium). In a system where molecules in the environment could potentially be washed away from the target area easily, in other words in a non-closed system, this would not be idea as our bacteria would have to be producing large amounts of ai-2 and would likely not be able to be supplemented by allowing ai-2 concentrations to build up prior to use.
This pushed us towards looking to non-bacterial coatings which may allow for higher initial concentrations, and to look at larger molecules such as QQ5 as this model indicated that the ability to move between cells solely by passive diffusion may not be appropriate to our intended application.

References

Berendsen, H. J. C., van der Spoel, D., van Drunen, R. GROMACS: A message-passing parallel molecular dynamics implementation. Comp. Phys. Comm. 91:43-56, 1995.
Lindahl, E., Hess, B., van der Spoel, D. Gromacs 3.0: A package for molecular simulation and trajectory analysis. J. Mol. Mod. 7:306-317, 2001.
Yang NJ, Hinner MJ. Getting across the cell membrane: an overview for small molecules, peptides, and proteins. Methods Mol Biol. 2015;1266:29-53. doi: 10.1007/978-1-4939-2272-7_3. PMID: 25560066; PMCID: PMC4891184.
. Jo, T. Kim, V.G. Iyer, and W. Im (2008) CHARMM-GUI: A Web-based Graphical User Interface for CHARMM. J. Comput. Chem. 29:1859-1865
I.D. Pogozheva, G.A. Armstrong, L. Kong, T.J. Hartnagel, C.A. Carpino, S.E. Gee, D.M. Picarello, A.S. Rubin, J. Lee, S. Park, A.L. Lomize, and W. Im (2022) Comparative Molecular Dynamics Simulation Studies of Realistic Eukaryotic, Prokaryotic, and Archaeal Membranes. J. Chem. Inf. Model. 62:1036-1051
V. Zoete, M. A. Cuendet, A. Grosdidier, O. Michielin, SwissParam, a Fast Force Field Generation Tool For Small Organic Molecules, J. Comput. Chem, 2011, 32(11), 2359-68. PMID: 21541964, DOI: 10.1002/jcc.21816

AHL DEGRADATION

Model

Modelling the degradation of AHL by one of our anti-biofilm molecules AHL lactonase is vital to understanding if our project is feasible. To do this we have used an adapted form of Michaelis Menten kinetics ^[1]:

$$v=\frac{k_{cat}\cdot [S]}{k_{m}+[S]}\cdot [E_{t}]$$

We have also added a production rate of AHL to turn our differential equation into:

$$v=\frac{k_{cat}\cdot [S]}{k_{m}+[S]}\cdot [E_{t}] + production$$

We have then assumed that the degradation of AHL lactonase is negligible. From this simple equation we can then use the following constants to determine the quantity of AHL lactonase we would need to degrade AHL in a meaningful way.

Constants

Constant	Value	Reference
K_cat	37.63 s^-1	[2]
K_m	0.00511 M	[2]
Production	0.167 nMs^-1	[3]

line
Using Copasi^[4] we then modelled this by using the inbuilt Michaelis Menten kinetics. We also modelled a ‘producer’ molecule with a very large concentration using the inbuilt mass action kinetics with a rate equal to the modelled production rate. Using Python we graphed a variety of different concentrations taken from our standard curve for our LCMS lab work. We also have a ‘negligible’ concentration of AHL, a concentration below which the Quorum Sensing capabilities of the bacteria are severely diminished and therefore biofilm formation may be affected.^[4].

Conclusion

Graph of AHL degradation showing three differening concentraions of AHL lactonase all reaching equilibriums as varying concentrations of AHL.

As we could see from this graph, we do achieve a plateau in our graph. From this we can then find a concentration for the lactonase that will result in an AHL concentration below our ‘negligible’ concentration. We can then ‘zoom’ into the graph to get a more clear vision of the ‘negligible concentration of AHL line.

Graph of AHL degradation showing two different concentrations of AHL lactonase both reaching plateaux, The ahl AHL-lactonase concentration 1.12e-5 Molar pleataus above the negligible concentration line and the AHL-lactonase concentration 1.0019e-4 pleateus below the negligible concetration line

From our graph we can see our ‘negligible’ concentration is between 1.12e-5 M and 1.0019e-4 M. Due to the rate of reaching the equilibrium being fast enough we could then assume that $\frac{\mathrm{d} [S]}{\mathrm{d} t}\approx 0$ for most of the time spent on the catheter. From that we can then plug in our negligible concentration to find our minimum concentration of AHL lactonase to disrupt biofilm formation, which comes out as 22.7 μM.

What future teams can do:

Model the degradation of AHL lactonase to find an initial concentration of AHL lactonase needed.

From this calculate the feasibility and cost of harvesting AHL lactonase for adding just the AHL lactonase

Calculate the production rates of the AHL lactonase needed by the L. plantarum to counteract the degradation of AHL lactonase and keep a concentration about 22.7 μM.

References

Park C. Visual interpretation of the meaning of kcat/km in enzyme kinetics. Journal of Chemical Education. 2022;99(7):2556–62. doi:10.1021/acs.jchemed.1c01268
Wang L-H, Weng L-X, Dong Y-H, Zhang L-H. Specificity and enzyme kinetics of the quorum-quenching N-acyl homoserine lactone lactonase (AHL-lactonase). Journal of Biological Chemistry. 2004;279(14):13645–51. doi:10.1074/jbc.m311194200
Simulate the Bacteria Quorum Sensing Process and Kill Switch [Internet]. iGEM; 2020 [cited 2023 Oct 10]. Available from: https://2020.igem.org/Team:Fudan/Model
Hoops S, Sahle S, Gauges R, Lee C, Pahle J, Simus N, et al. COPASI—a complex pathway simulator. Bioinformatics. 2006;22(24):3067–74. doi:10.1093/bioinformatics/btl485
Nilsson P, Olofsson A, Fagerlind M, Fagerström T, Rice S, Kjelleberg S, et al. Kinetics of the AHL regulatory system in a model biofilm system: How many bacteria constitute a “quorum”? Journal of Molecular Biology. 2001;309(3):631–40. doi:10.1006/jmbi.2001.4697

MODELLING

Alphafold

Introduction

Model

Returning to the Literature

References

RIP-RAP-TRAP

Introduction

Model

RIP RAP TRAP mechanism

Agr operon

Degradation and production rates

Constants

Conclusion

Concerns

References

DPD

Introduction

Model

Differential Equations

Constants

Conclusion

References

2D DIFFUSION

Introduction

Model

Conclusion

3D DIFFUSION

Introduction

Model

Conclusion

References

AHL DEGRADATION

Model

Constants

Conclusion

References