Difference between revisions of "Team:IISER-Tirupati India/Model"

Introduction

Natural systems are highly complicated to be studied with just experiments. Hence mathematical models are built using the existing data and information to predict and extrapolate the outcomes. This subsequently gives us an insight into the working of the system without experimental probing. In this section we attempt to examine various theoretical aspects of our projects. This helps us give pointers for future experiments and implementations. Here we will be exploring the ways in which the genetically modified bacteria of our interest would be introduced to the system and its interaction with the environment. In this cascade study, we find to what extent our GMC proliferates and then compute the minimum theoretical efficacy of our process.

The different facets of our project will include:

1. GMC delivery
2. GMC growth and colonisation
3. Protease production
4. Diffusion and ovum hardening
5. Kill switch for reversibility
6. Kill switch for biosafety

Delivery & Colonisation(Module 1)

Introduction: The journey of GMC (Genetically Modified Commensal) begins.

The GMC i.e modified lactobacillus acidophilus needs to be delivered to the Ampulla of the fallopian tube where they shall colonise as long as contraception is needed. Although the proposed bacteria is Lactobacillus Acidophilus, the model organism for this project is Bacillus subtilis. Once delivered, the bacteria stick to the mucous membrane and colonise with the help of nutrients available through the oviductal fluid. They constantly produce ovastacin until progesterone levels rise post ovulation, hence three days pre and post ovulation the GMC act on hardening the ovum ZP2. This module deals with the transport and growth of the bacteria in the fallopian tube.  

Part A: Delivery of GMC 

Overview:

Delivery is an essential part of our project as it determines the type of user  we reach. We tried to develop multiple options to deliver GMC in a user-friendly way with minimum invasion. Minimum invasion means it should not colonize the whole reproductive tract or reach the ovaries. The device should ensure bacterial colonization specifically in the ampulla region of the fallopian tube.

There were several limitations and constraints we faced such  as immotility of the GMC, strictly selective region for colonisation, the estimated time taken for the whole process etc. Several ways were attempteded (like Direct injection of a fluid from ostia, Cell encapsulation, etc. from the available data provided in [1]), which should be within our desirable limit of the constraints, however all of these efforts were fruitless . Eventually, on  further discussions  with an IVF specialist Dr.Sidra Khot, gynecologists Dr.Prameela Menon and Dr. Frances Grimstad,  we came to the conclusion that hysteroscopic techniques would be the best for our purpose[4] . This method gives us an advantage by delivering the bacteria directly into the ampulla region. 

Procedure 

GMC will be delivered directly to the ampulla by a technique called Hysteroscopy [4]. It’s done with medical assistance and requires constant X-Ray monitoring to ensure the catheter is inserted without any issues. The reason is that the Ostia and the Fallopian tube are dynamic structures and that there is no chemical or significant physical difference between the ampulla and the other components of the Fallopian tube.

As advised and recommended by the IVF specialist. The diameter of the Hysteroscope is 2.9 mm and it further narrows for the part that enters the Fallopian Tube and the volume of the Hysteroscope is 2-3mL. We’ll use air as our pushing entity to inject 0.1 mL oil or a saline solution containing our GMC. Oil is USG opaque which helps in guidance for the injection of the droplet
The bacteria is sensitive to light so any transparent part of the equipment is covered or tinted. First light is used for guidance and then the bacteria is injected in the absence of light.

 

A solution of GMC of a specific concentration is prepared. Then the catheter is inserted under medical guidance up to the ampulla. Then an injector is projected with narrow long incisions on both sides of it, which ensures that the GMC forms a ring along the circumference of the tube. Ovum is now 360° surrounded by the GMC.

The introduced bacteria on the wall produces our desired protease-ovastacin denoted diagrammatically as the red circle in the image below. The ovum is the green circle (size is greatly exaggerated) and only an area segment is shared between the two circles. I.e, not all ovastacin produced is in contact with the ovum and not all parts of the ovum is affected by the ovastacin. Those that matter are represented by the intersecting area between the two circles.

The GMC Lactobacillus acidophilus has pili which expectantly forms hydrogen bonds at the tube circumference with mucin found in mucus. The GMC is now adhered to the walls and multiplies once it adjusts to the introduced environment (lag phase)[6,7]

Future Aspects: 

Since we aim to make our final product more accessible, the hysteroscopic technique, we speculate, is expensive (mostly in private hospitals). We also considered using hydrogels for the delivery of bacteria at a pH-specific region.
Why is this system compatible with the delivery of bacteria in the ampulla of the Fallopian tube region?

We found out that the pH in the oviductal ampulla region is close to 8.0.That’s why we considered using hydrogels for bacterial delivery, which swells up at a pH range of 7.8 to 8.0.[2]

Amongst the stimuli-responsive hydrogels, pH-sensitive hydrogels are the most studied hydrogels. The rate at which hydrogels respond depends upon their size, shape, cross-linking density, number of ionic groups, and composition, which can be tailored by varying these factors. The response rate increases with increasing pore size and number of ionic groups and decreasing their size and cross-linking density.

For a delivery in a basic medium, we planned to use anionic hydrogels, such as carboxymethyl chitosan, which swell at higher pH (basic medium) due to ionization of the acidic groups[3][5]. As a result, the ionized negatively charged pendant groups on the polymer chains cause repulsion leading to swelling. This property of hydrogels can be exploited for GMC delivery at pH 7.4 in the ampulla of the fallopian tube. 

However, this idea of cell encapsulation by hydrogels was scrapped as it provides no additional benefit as compared to the aforementioned Hysteroscopic method under USG guidance.

References :

[1] Soriano-Úbeda, C., García-Vázquez, F., Romero-Aguirregomezcorta, J. et al. Improving porcine in vitro fertilization output by simulating the oviductal environment. Sci Rep 7, 43616 (2017). https://doi.org/10.1038/srep43616

[2] Human Reproduction Update, Volume 24, Issue 1, January-February 2018, Pages 15–34, https://doi.org/10.1093/humupd/dmx028

[3] Carboxymethyl Chitosan and Its Hydrophobically Modified Derivative as H-Switchable Emulsifiers Langmuir 2018 Feb 27;34(8):2800-2806.

doi: 10.1021/acs.langmuir.7b03959. Epub 2018 Feb 13.

[4] Howard W. Jones, John A. Rock - Te Linde’s Operative Gynecology-LWW (2015) from pg:365

[5] Hydrogels and Their Applications in Targeted Drug Delivery

Radhika Narayanaswamy and Vladimir P. Torchilin Molecules. 2019 Feb; 24(3): 603.

Published online 2019 Feb 8. doi: 10.3390/molecules24030603

[6] Buck, B. L., Altermann, E., Svingerud, T., & Klaenhammer, T. R. (2005). Functional analysis of putative adhesion factors in Lactobacillus acidophilus NCFM. Applied and environmental microbiology, 71(12), 8344–8351. https://doi.org/10.1128/AEM.71.12.8344-8351.2005

[7] Jugal Kishore Das, Rajani Kanta Mahapatra, Shubhransu Patro, Chandan Goswami, Mrutyunjay Suar, Lactobacillus acidophilus binds to MUC3 component of cultured intestinal epithelial cells with highest affinity, FEMS Microbiology Letters, Volume 363, Issue 8, April 2016, fnw050, https://doi.org/10.1093/femsle/fnw050

[8] Boris, S., Suárez, J. E., Vázquez, F., & Barbés, C. (1998). Adherence of human vaginal lactobacilli to vaginal epithelial cells and interaction with uropathogens. Infection and immunity, 66(5), 1985–1989. https://doi.org/10.1128/IAI.66.5.1985-1989.1998

Part B: Colonization

Overview 

Growth kinetics is crucial to find out the behaviour of the bacterial population over time. Growth can be affected by external factors such as temperature, composition & concentration of nutrients, pH, competition with other strains and species, and different physiological factors such as growth factor proteins[1][2]. Bacteria are capable of producing useful biomolecules, which humans utilize for different purposes. This phenomenon has applications, from producing delicious Italian wine to getting antibiotics which saves millions of lives every year. We shall use this simple formula to reach the goals of our project. To have a qualitative understanding of the protein production by the Genetically Modified Commensal (GMC), we need to have a theoretical approach. We need to understand how we can calculate the growth of GMCs [3]. This, in turn, will help us figure out protein production. This detailed study of growth kinetics will help us calculate the initial inoculation required for the production of a protein that we need.

For a better understanding, we considered two cases. The first comprises the study of the growth of Bacillus subtilis in normal petri dish conditions. This is the known environment and we can have the data through an experimental growth curve. The second case of study is the continuous culture model where we study the growth of GMC in presence of Lactobacillus acidophilus..   

Assumptions

1. Inhibitions due to toxins produced by other bacteria are ignored
2. We’ll focus only on one single substrate (presumably Glucose) and assume that no other bacteria utilises it [5]

Model Development (Growth Models)

There’re formulas that are available to model the growth of interacting populations. We chose go with Continuous Haldane-Logistic growth model as they’re more accurate [4] and describes more interactions (other growth models were nevertheless numerical calculated but their results weren’t desirable to us)

where,

• $X(t)\rightarrow$ Population number of the $i^{th}$ population
• $\mu_i (t)\rightarrow$ Growth rate of the $i^{th}$ population
• $\mu_{m_i} (t)\rightarrow$ Maximum growth rate of the $ i^{th}$ population
• $S(t)\rightarrow$ Substrate concentration
• $S_F\rightarrow$ Concentration of Substrate in Feed stream of the bioreactor
• $Y_{{x_i}/{s}}(t)\rightarrow$ Yield coefficient of the population
• $D\rightarrow$ Dilution rate of the bioreactor
• $K_{s_i} \rightarrow$ Half velocity/Saturation constant-the value of $S(t)$ when $\frac{\mu}{\mu_i}=0.5$ for the $ i^{th}$ population
• $K_{f_i} \rightarrow$ Inhibition constant of the $i^{th}$ population
• $X_{m_i} \rightarrow$ Maximum population of $i^{th}$ population

Mathematica code

Download

These are empirical parameters that can only be determined from the experiment because they change with environment. 

Modelling the two populations as: "Unmodified Lactobacillus acidophilus" (Variant-1) and "Introduced GMC"(Variant-2).

Interpretation:

The values of certain parameters were not available through literature, which led to building three cases based on different values for the parameters. 

This is a good example of Chaos Theory meaning that there are several parameters and initial conditions that can drastically affect the resultant growth curve.

 

References:

[1] Liu, Shijie (2017). Bioprocess Engineering || How Cells Grow., 629–697. doi:10.1016/B978-0-444-63783-3.00011-3.

[2] Stewart, C. E.; Rotwein, P. (1996). Growth, differentiation, and survival: multiple physiological functions for insulin-like growth factors. Physiological Reviews, 76(4), 1005–1026. doi:10.1152/physrev.1996.76.4.1005 

[3] Trivier, D., Davril, M., Houdret, N., & Courcol, R. . (1995). Influence of iron depletion on growth kinetics, siderophore production, and protein expression of Staphylococcus aureus. FEMS Microbiology Letters, 127(3), 195–200. doi:10.1111/j.1574-6968.1995.tb07473.x 

[4] Mpho Muloiwaa,, Stephen Nyende-Byakikab, Megersa Dinka Comparison of unstructured kinetic bacterial growth models. South African Journal of Chemical EngineeringVolume 33, July 2020, Pages 141-150

[5] Utilization of sugars by Lactobacillus acidophilus strains D Srinivas , B K Mital, S K Garg Int J Food Microbiol 1990 Jan;10(1):51-7. doi: 10.1016/0168-1605(90)90007-r.

3. Protease production

1. Total amount of ovastacin required: 

Overview 

Well known for extracellular protease production[1], Bacillus subtilis is our model organism for the proof of concept experiments. Moreover, it is a gram-positive bacteria which even if not too close but is closer than e coli (gram-negative) to our proposed bacteria Lactobacillus acidophilus. This module deals with the whole mechanism of action ideally the GMC must follow for contraception to be attained. To know how this module developed to what it is, please refer to the engineering success;

Let's create the flow, you must know by now that we plan to produce the protease known as ovastacin, an indigenous protease[2] to the human ovum. It is known to cause Zona pellucida hardening naturally used by the ovum to prevent polyspermy[3]. To see the mechanism of zona pellucida hardening click here (goes to project overview where it is explained). 

Assumptions:

1. One active ovastacin cleaves one ZP2 glycoprotein present in the Zona pellucida matrix. 
2. Radius of ovum and thickness of zona pellucida is taken as the average of the highest and lowest values. 
3. Every glycoprotein (ZP1,2,3,4) is a sphere of size equal to that of ZP2 because their size differences are insignificant. 
4. The four spheres together are approximated to form a cuboid as shown in the figure.

Model Development:

To get the number of molecules of ovastacin needed, the number of ZP2 glycoproteins present on the surface of the ovum were calculated.

Calculations and Simulations: 

Assuming a cuboid with all spheres of size same as ZP2 to get the volume of one unit = 3.1 $\times$ 10^-6 μm³ 

Volume occupied by whole ZP matrix = 12.4 $\times$ 10⁵ μm³ 

Calculated the number of ZP2 glycoproteins present in the ZP matrix = 3.92 $\times$ 10¹¹ molecules

Assuming 1 ovastacin cleaves 1 ZP2

No. of ovastacin = No. of ZP2 = 12.4 $\times$ 10⁵ / 3.1 $\times$ 10^-6 = 3.92 $\times$ 10¹¹ molecules 

Interpretation: 

In order to cause complete hardening, 3.92 $\times$ 10¹¹ a number of molecules are required to reach the ovum. The next question that arises is how much is produced and how much of produced ovastacin will reach the ovum.  

2. Production of ovastacin through GMC

Overview: 

To determine how much ovastacin can be produced, the genetic circuit (click here) was constructed and studied. Main concern of the circuit was to ensure proper expression of the protease enough to cause hardening of the ovum right after ovulation. The genetic circuit was built such that ovastacin shall be produced as long as the progesterone levels lie low i.e. before ovulation. Since overproduction is not intended, a repressor of ovastacin is used such that it’s levels start writing right after ovulation. The repressor P22 represses ovastacin production by binding to its operator as a dimer. The model is developed using the interactions among these three cassettes.

Assumptions: 

1. Bacteria are in the stationary phase.
2. Protein production doesn’t reach saturation i.e. no inclusion bodies.
3. Concentration is in nanomolar and time in days.
4. The progesterone serum levels are considered for the model because the day wise data for tubal fluid/ follicular fluid was not available.
5. 1 SRTF binds to 1 Operator, All SRTF in cells are bound at t=0.
6. All the ovastacin produced in the cell is transported outside the cell using the TAT pathway.

Model Development: 

Using the assumptions stated above the ODEs were developed based on the genetic circuit. The external source to the circuit namely progesterone varies over the period of a month in a similar way every time. Daily values of progesterone were used to obtain it’s graph over a month through curve fitting and the best fitting function was used for the ODEs.

The allosteric transcription factor SRTF1 binds to DNA in the absence of progesterone. How strongly it binds, depends on it’s KD (dissociation constant) value with progesterone binding. Srtf binds to P22 promoter as a dimer, once the hormone comes inside the cell the repression of P22 stops. Srtf and progesterone from complex while P22 gets produced. P22 then binds as a dimer to the operator of the promoter for ovastacin expression to repress it. Hence when there is a progesterone surge right after ovulation P22 shall suppress the ovastacin production. While getting the graphs for this setup, the production strengths of these proteins had to be varied compared to the chosen parts in order to achieve the proper cycle. These changes can be incorporated as future aspects to this project. For the purpose of studying such a system, we take an example of a promoter producing a protein, and write the following derivation with,

• mRNA = [m]
• RNA polymerase = [R_p]
• Protein = [p]
• Promoter = [p_o]
• [ribosome] = [r]

A) For Derivation

B) For Model

Similarly, equations for all the proteins can be written in the same way taking into consideration the interactions of repressors like Srtf and P22 with the DNA using Hill’s equations. Taking,

• [progesterone] = [Pr]
• [SRTF1]=[S]
• [Ovastacin] = [O]
• [Fetuin B] = [FB]
• [P22] = [P22]
• [Ovastacin]=[O]
• [SRTF-1]=[S]

We write,

Calculation and Simulation: 

Graphs

Taking some approximate values for the parameters and constants to see the expression of ovastacin with time we got the following graphs using ode solver MATLAB. 

Matlab Code

Download

Interpretation: 

The graph obtained shows that ovastacin can be produced from 11th to 17th day of the menstrual cycle where ovulation is most likely to take place on Day 14. A total of 152087 nanomolar of ovastacin can be generated. Once ovastacin is secreted it has to reach the zona pellucida of the ovum, while doing so it shall encounter various roadblocks like dispersion in the tubal fluid flow, action of other proteins, fetuin B and its own degradation rate. These factors other than fetuin B which was already considered here shall be discussed further.

References: 

[1] Production, optimization and partial purification of protease from Bacillus subtilisGaurav Panta, Anil Prakasha, J.V.P .Pavania, Sayantan Beraa, G.V.N.S. Devirama, Ajay Kumara, Mitali Panchpurib, Ravi Gyana Prasunaa.

[2] ASTL l [Astacin-like metalloendopeptidase] UniProtKB - Q6HA08 (ASTL_HUMAN) https://www.uniprot.org/uniprot/Q6HA08 

[3] Ovastacin, a cortical granule protease, cleaves ZP2 in the zona pellucida to prevent polyspermy Anna D. Burkart, Bo Xiong, Boris Baibakov, Maria Jiménez-Movilla, and Jurrien Dean. doi: 10.1083/jcb.201112094

[4] Does zona pellucida thickness influence the fertilization rate? E. Bertrand1 , M.Van Den Bergh and Y.Englert

[5] Characterization of human zona pellucida glycoproteins A R Bauskin1, D R Franken, U Eberspaecher, P Donner DOI: 10.1093/molehr/5.6.534

[11]Daily plasma progesterone levels during the menstrual cycle C. A. WOOLEVER, M.D. Denver, Colorado

4. Journey of Ovastacin (Diffusion) 

Overview

From the papers, it’s understood that the ovum is propelled by the ciliary motion away from the uterus and that means the ovum is in contact with the wall. The size ampulla region of the Fallopian tube (2.5 mm ≤ radius ≤ 5 mm ) is of much larger than the radius of the ovum (61.7μm) A molecule under the influence of Brownian motion move according to the equation

• For 2-D, \begin{equation}\tag{3.1} \left\langle r^2\right\rangle=4Dt \end{equation}
• For 3-D, \begin{equation}\tag{3.2} \left\langle r^2\right\rangle=6Dt \end{equation}

Where,

• Diffusion Coefficent \begin{equation}\tag{3.3}D=\frac{k_BT}{6\pi \eta r}\end{equation}


• t → Time elapsed

Using,

1. k_B = 1.38064852 ×10^-23 m² kg s^-2 K^-1 [Boltzmann constant]
2. η = 0.799 Pa·s [Viscosity of Oviductal Fluid] [1]
3. T = 35.5 + 273.15 = 308.65 K [Temperature of Oviduct]
4. r = 22 kDa = 2.43 nm [Radius of ovastacin molecule]
5. a = 61.7 μm [Radius of the ovum] [6]
6. l=18.9 μm [Thickness of ZP2 layer]
7. s = 5 mm [Radius of the ampulla region of the oviduct]
8. N₀ = 3.92 ×10¹¹ molecules = 6.51 $\times 10^{-4}$ nanomoles [Number of ovastacin to react with all ZP2]

For, Ovastacin D=1.16442 ×10^-13 m² s^-1

Assumptions

1. 100 % Effusion permeability for ovastacin through ovum
2. The surface of the ovum and the oviduct is assumed to be smooth
3. The mean squared radial distance of molecules is taken to be the exact value of the position \begin{equation*}r^2(t) \approx \left\langle r^2\right\rangle=6Dt\end{equation*}
4. Ovum is so small compared to the diameter of the oviduct that the ovastacin front is assumed to be a Euclidean plane
5. Losses due to getting trapped between Cilia, Reaction with Fetuin B is ignored.
6. When rolling all parts of the ovum are assumed to receive uniform exposure to the ovastacin.
7. No slip condition for rolling.

Model Development

From the image below, ovum is much smaller than the Fallopian tube when a small section is considered.

The introduced bacteria colonises the blue circle (around the circumference of the oviductal cross-section). The average time taken by ovastacin from the opposite end of the circle to reach the ovum is

\begin{equation}\tag{3.4}t=\frac{4(s-a)^2}{6D}=139622117.976 \:s > 4 \; years \end{equation}

Definition of t_100%

The minimum amount of time is takes to completely engulf (react with 100% of the ZP) the ovum with Ovastacin ($t_{100\%}$) = (Time taken to travel ’2a’ [Diameter])+(Time taken to travel a’ [Down the radius])

\begin{equation}\tag{3.5}\frac{4a^2+a^2}{6D}=27244.549025\:s>7.5\:hrs.\end{equation}

Ratio of ovastacin reaching and absorbed

Assuming that all the ovastacin that reaches the ovum is absorbed and it’s

evenly distributed across the space (Constant concentration ρ). 

The ratio P₁ is defined as the ratio between the number of ovastacin molecules produced overall with the number of ovastacin molecules absorbed at given time (t).

\begin{equation}\tag{3.6} P_1(t)=\frac{No.\;of\;ovastacin\;absorbed}{No.\;of\;ovastacin\;produced\;in\;total}=\frac{\rho V_{ovum}-\rho V_{intersection}(t)}{\rho V_{produced}(t)}=\frac{V_{ovum}-V_{intersection}(t)}{V_{produced}(t)}\end{equation}

Percentage of ovum (G) that’s reacted bu ovastacin can be given by

\begin{equation}\tag{3.7}G=\frac{V_{react}}{V_{ovum}}=\frac{V_{ovum}-V_{intersection}}{V_{ovum}}=1-\frac{V_{intersection}}{V_{ovum}}\end{equation}

Henceforth a line source is considered for the small section

Approximating a<<s, the red region bounded by the sphere and the black line is chosen to be the reacted region as shown in the figure whose formula is given by

\begin{equation}\tag{3.8} V_{react}=\frac{6\pi Dt(3a-\sqrt{6Dt})}{3} \end{equation}

Hence calculating,

\begin{equation}\tag{3.9} G(t)=\frac{6\pi Dt(3a-\sqrt{6Dt})}{3 \times \frac{4}{3}\pi a^3}=\frac{3 Dt(3a-\sqrt{6Dt})}{2a^3} \end{equation}

Definition of t_70%

For the ovum to be completely hardened G ≥70%. Hence

\begin{equation}\tag{3.10} 0.7=\frac{3 Dt_{min}(3R-\sqrt{6Dt_{min}})}{2a^3} \end{equation}

We obtain that t₇₀ = 8836.846 s (approximately 2 hrs 30 mins).
The volume containing the ovastacin is a cylindrical shell of length ‘h’, thickness ‘\(\sqrt{6Dt}\)’ and outer radius ‘s’ \begin{equation}\tag{3.11} V_{produced}(t)=\pi^2h\left[s^2-(s-\sqrt{6Dt})^2\right]\end{equation}
Redefining P₁ 

\begin{equation}\tag{3.12} P_1(t)= \begin{cases} \frac{2Dt(3a-\sqrt{6Dt})}{h\left(s^2-\left(s-\sqrt{6Dt}\right)^2\right)} ,& t<\frac{9a^2}{6D} \\ \frac{\frac{4}{3}\pi a^3}{V_{produced}(t)} ,& t \geq \frac{9a^2}{6D} \end{cases} \end{equation}

where,
$\frac{9a^2}{6D}=49040.19 s \approx$ 13 hr 37 min 20 sec

Ovastacin degradation with time:

These losses (P₂) are already included in the protease production module. Hence, from the graph we have

Ovastacin which gets washed away by oviductal fluid flow

\begin{equation}\tag{3.13} Q(Flux)=\frac{V}{t}=A\times v_{fluid}\end{equation}

\begin{equation}\tag{3.14} N_{washed}(t)=C_{ovastacin}V_{washed}=C_{ovastacin}\times A\times v_{fluid} \times t \end{equation}

\begin{equation}\tag{3.15} C_{ovastacin} = \frac{N(t)}{V_{produced}}\end{equation}

\begin{equation}\tag{3.16} N_{washed}(t)=C_{ovastacin}V_{washed}=\frac{N(t) \pi [s^2-(s-\sqrt{6Dt})^2] v_{fluid}t}{V_{produced}}=\frac{N(t) v_{fluid}t}{h} \end{equation}

Hence,

\begin{equation}\tag{3.17} P_3(t)=\frac{N_{unwashed}(t)}{N(t)}=\frac{N(t)-N_{washed}(t)}{N(t)}=1-\frac{v_{fluid}t}{h} \end{equation}

Using,

• $V_{fluid}$ = 0.07 μm s^-1[2]
• a = 61.7 μm [Radius of the ovum]

Calculation and Simulation:

Ovulation takes place at the 14^th day. It takes 1 day [4] for the ovum to travel from the ovary to the ampulla. The ovum remains stationary for 3 days [4] in the ampulla which is a larger duration than t_70\% or t_100%. It’s also to be understood that the ovum can stop at any part of the ampulla and can last for upto 80 [4] or 144 hours [7] hence we’ll colonise the whole ampulla instead of a specific strip (h=5 or 8 cm). From this,

\begin{equation}\tag{3.18}N_{consumed}(t)G(t)=N_{produced}(t)P_1(t)P_2(t)P_3(t)\end{equation}

Let the time taken for the process be t_f, so after t_f ensuring that 100% of the ovum is cleaved ($G( t_f)=1$). and $N_{consumed}(t_f)=N_0= 6.51 \times 10^{-4} nanomoles $ and degradation loss already accounted for in the graph we get

\begin{equation}\tag{3.19}N_0=B\times[Ovastacin]\times V_{produced}(t_f)\times\frac{\frac{4}{3}\pi a^3}{V_{produced}(t_f)}\times\left(1-\frac{v_ft_f}{h} \right )\end{equation}

where, B is the number of minimum final GMC present. Rearranging

\begin{equation}\tag{3.20}B=\frac{N_0}{[Ovastacin]\times \frac{4}{3}\pi a^3 \times P_3(t_f)}\end{equation}

For h=5 cm,

For h=8 cm,

Interpretation

The major loss factors of ovastacin so far were mentioned in the sections above. These’re the simplest factors or it has been simplified hence a lot of assumptions have been taken. Combining those loss factors gives the overall equations as.

\begin{equation}\tag{3.21}N_{consumed}(t)G(t)=N_{produced}(t)P_1(t)P_2(t)P_3(t)\end{equation}$

Where,

\begin{equation}\tag{3.22}N_{produced}(t)=\int_{0}^{t} kX(t) \; dt \end{equation}

• X(t) is the population of the GMC
• k → Production rate of ovastacin per bacterium

The average minimum production rate can then be written as

\begin{equation}\tag{3.23} \left \langle k \right \rangle =\frac{N_{produced}(t_f)}{t_f}\end{equation}

The time taken for the process (t_f) is variable for our convenience. From this we can find the amount of inoculum needed.

Rotation factor of ovum

According to literature, the ovum stays within the ampulla for 3 days. So assuming that, with no slipping the ovum rolls throughout the length of the ampulla (L). The angular velocity (ω) can be calculated as

\begin{equation}\tag{3.24}\omega =\frac{L}{3\;days\times Radius\;of\;ovum (a) } \end{equation}

For a very small δt it can be approximated as shown in the image below.

Using φ = ωt ignoring the origin shift, the equation of the blue spiral can be interpreted as below

\begin{equation}\tag{3.25}x(\varphi)=\left(a-\sqrt{\frac{6D\varphi}{\omega}}\right)\cos{\varphi}\end{equation}

\begin{equation}\tag{3.26}y(\varphi)=\left(a-\sqrt{\frac{6D\varphi}{\omega}}\right)\sin{\varphi}\end{equation}

The tangent equation for any angle φ on the blue circle is given as

\begin{equation}\tag{3.27}(y-y(\varphi))=m(\varphi)(x-x(\varphi))=\frac{dy(\varphi)}{dx(\varphi)}(x-x(\varphi))\end{equation}

By 2π rotation almost all the ZP layer is covered and obviously at 4π all of  ovastacin (so far produced) is in contact with the ZP, but it must be kept in mind that the "circle" is a sphere and the tangent "line" is a plane.

Let $\varphi=\varphi_{min}$ which is the smallest angle needed to completely enclose the ZP-layer. $\varphi_{min}$ was solved by Mathematica and from a range of values the smallest value of $\varphi_{min}$ between $2\pi \leq \varphi_{min} \leq 4\pi $ is the correct one. The values thus obtained are as follows.

Now it needs to be kept in mind that the real angular velocity ($\omega$) is significantly larger than the one calculated above since the ovum is actually stationary for most of its time. So it’s possible that the ovum gets 100% exposure in the first day after ovulation itself.

Mathematica code

Download

References

[1] Numerical simulation of embryo transfer: how the viscosity of transferred medium affects the transport of embryos Dali Ding, Weiping Shi & Yang ShiTheoretical Biology and Medical Modelling volume 15, Article number: 20 (2018)

[2] Mechanics of transport of the ovum in the oviduct Sneh Anand & Sujoy K. Guha Medical and Biological Engineering and Computing volume 16, Article number: 256 (1978)

[3] Ghazal, S, Kulp Makarov, J, et al, Glob. libr. women's med., (ISSN: 1756-2228) 2014; DOI 10.3843/GLOWM.10317

[4] Reference Module in Biomedical Sciences. Chapter Gamete Transport

[5] Physiology: Human tubal fluid: formation and composition during vascular perfusion of the Fallopian tube C.J. Dickens, S.D. Maguiness, M.T. Comer, A. Palmer, A.J. Rutherford, H.J. Leese Human Reproduction, Volume 10, Issue 3, March 1995, Pages 505–508, https://doi.org/10.1093/oxfordjournals.humrep.a135978

[6] Influential effect of age on oocyte morphometry, fertilization rate and embryo development following IVF in mice Masoomeh Mohammadzadeh, Farzaneh Fesahat, Arezoo Khoradmehr, Mohammad Ali Khalili https://doi.org/10.1016/j.mefs.2017.09.006

[7] Studies on the duration of egg transport by the human oviduct. II. Ovum location at various intervals following luteinizing hormone peak H B Croxatto, M E Ortiz, S Díaz, R Hess, J Balmaceda, H D Croxatto Am J Obstet Gynecol. 1978 Nov 15;132(6):629-34.doi: 10.1016/0002-9378(78)90854-2

5. Population for required Ovastacin

Overview:

Until now the growth model, ovastacin’s production, and diffusion have been studied. This part aims to get an idea of how much the bacteria shall colonise and how much inoculum needs to be injected. The fallopian tube is a system with multiple species: the number of each individual species and the total number of species will oscillate about a certain equilibrium position. First, the final population required is figured out using the diffusion and production models of ovastacin. Then using the following assumptions a system with multiple species is constructed and inoculum calculated.

Assumptions:

1. This system is at a stable equilibrium meaning any perturbation of the system will bring it back to the equilibrium. The equilibrium being the carrying capacity.
2. The GMC does not have a survival advantage or disadvantage against normal Lactobacillus.
3. The inner walls of the fallopian tube are smooth.

Model Development:

• $M \rightarrow$ Carrying capacity of the system for all the commensals.
• $N_0 \rightarrow$ Initial amount of Normal Lactobacillus Acidophilus
• $N_f \rightarrow$ Final amount of Lactobacillus Acidophilus
• $I \rightarrow$ Inoculated/introduced amount of GMC
• $a \rightarrow$ Final amount of GMC

The derivation is based on the steady state achieved by the population after a long time. Since the oviduct is highly competitive we assume that the average proportion of each species is constant. So the ratios are given below as

\begin{equation}\tag{4.1}\chi_1=\frac{N_0}{M}=\frac{N_f+a}{M}\end{equation}

\begin{equation}\tag{4.2}\chi_2=\frac{I}{N_0}=\frac{a}{N_f}\end{equation}

Assuming the survival factors of GMO and Lactobacillus are the same we continue by cancelling ‘M’ on both sides and substituting for ‘Nf’, Isolating ‘I’ we get,

\begin{equation}\tag{4.3}I=\frac{aN_0}{N_0-a}\end{equation}

Calculations and Simulations:

By, [1] the microbe count is >10 ⁸ CFU/mL of oviductal fluid.

In, [2] 1-5% is Lactobacillus acidophilus. Which gives us 10 ⁶ to 5 × 10 ⁶ CFU/mL

Treating the ampulla as a truncated cone of radii 10 mm and 5 mm of length 5 cm to 8 cm [3]. The volume of the ampulla region of the oviduct is 9.16 to 14.66 mL which means there’s a minimum of 0.92 × 10 ⁷ to 7.3 × 10 ⁷ CFU of Lactobacillus acidophilus in the ampulla.

This gives us ‘I’ to be 6835 to 6830 CFU.

Let the radius of the droplet (r_drop) be 0.75 cm. Hence Volume of droplet 1.76 mL (0.52 to 4.188 mL).

Now calculating the number of transfers required for the whole ampulla=$\frac{L}{2r_{drop}}$, where L is the length of the ampulla which is 5 or 8 cm.

The number of such transfers are hence, 4 or 6.

The concentration thus prepared is $\left(\frac{I}{Number\; of\; transfer\;\; \times\;\; Volume \;of\; droplet}\right)$ 644.165 to 966.955 CFU/mL.

To prevent possible blockage saline solution can be used although it was assured to us by the IVF specialist that it’s not likely to happen

If it’s a 0.1 mL droplet ($r_{drop}=0.288$ cm ) then 9 to 14 transfers would be required and 4882.14 to 7588.89 CFU/mL solution would be prepared.

Interpretation

Based on current assumptions, 644.165 to 966.955 CFU/mL inoculum concentration of bacterial fluid is needed to be injected through 4 to 6 transfers.

References

[1] Damage to oviduct organ cultures by Gardnerella vaginalis,Species specificity of attachment and damage to oviduct mucosa by Neisseria gonorrhoeae

[2] The fallopian tube microbiome: implications for reproductive health

[3] Ghazal, S, Kulp Makarov, J, et al, Glob. libr. women's med., (ISSN: 1756-2228) 2014; DOI 10.3843/GLOWM.10317

6. Protein Modelling

Introduction:

In the fallopian tube, upon fertilization, there’s a release of a metallo-protease ovastacin from the cortical granules of the oocyte into the perivitelline space [1]. It interacts with the outer layer of the oocyte leading to remodelling of the Zona Pellucida (ZP), the glycoprotein matrix which forms the outer layer of the oocyte, by cleavage of the Zona Pellucida 2 (ZP2) glycoprotein component of ZP, at a specific site [1]. This cleavage prevents sperm binding, renders the ZP impermeable and thus blocks polyspermy [1]. 

We took advantage of this specific protein-protein interaction to design our contraceptive and decided to produce recombinant ovastacin which can cleave ZP2 prior to fertilization thus cloaking the ovum from sperm. We studied the structure of both the proteins from literature and bioinformatic tools as mentioned further to understand this interaction that would form the basis of our contraceptive. These tools gave us a clearer picture of the structure and stability of the recombinant proteins which helped us proceed further with our design.

Ovastacin:

Ovastacin (encoded by ASTL gene) is a member of the astacin family and the metzincin superfamily of metalloproteases. Translation of ASTL gene gives ovastacin in an inactive zymogen (pro-ovastacin) form, including a signal peptide, a pro-domain, catalytic protease domain and C-terminal domain of unknown structure and function [1]. The pro-domain of ovastacin has an aspartate residue to block the zinc ion that is involved inactivity of the protease and, thus, removal of the pro-domain is required to gain activity [1].

Biochemical studies performed with crayfish ovastacin gives an overview of the activation of the protease from zymogen (inactive form) to its mature form [1]. The proastacin activation in the crayfish is a two-step process involving the removal and degradation of the prosegment [2]. In the first activation step, crayfish trypsin protease performs initial cleavages to render a series of premature astacin variants. In the second step, these variants, which are catalytically active, produce subsequent cleavages.

However, the activation process for human ovastacin hasn’t been studied in detail due to unavailability of samples and ethical issues associated with study of human ovum and zygote at different stages of development. Our data mining processes didn’t yield any sequence of active or mature forms of ovastacin. Hence, we decided to use the peptidase domain that is the ‘Peptidase M12A’ domain of pro-ovastacin which has the predicted active site residues for members of astacin family in its HEXXH motif, as our recombinant protein sequence [2].

Stage 1:

We then went on to predict the structure of this domain since there’s no crystal structure of mature ovastacin available. We used I-TASSER (Iterative Threading ASSEmbly Refinement), a platform for automated protein structure and function prediction based on the sequence to structure to function pattern to get a structure from the sequence that we considered to use and express recombinant protein from bacteria [3].

The following is the result that we received from the server.
The native protein sequence was obtained from pro-ovastacin sequence in uniprot (ID-Q6HA08) 

RLLSAASNKWPMGGSGVVEVPFLLSSKYDEPSRQVILEALAEFERSTCIRFVTYQDQRDFISIIPMYGCFSSVGRSGGMQV VSLAPTCLQKGRGIVLHELMHVLGFWHEHTRADRDRYIRVNWNEILPGFEINFIKSRSSNMLTPYDYSSVMHYGRLAF SRRGLPTITPLWAPSVHIGQRWNLSASDITRVLKLYGCS

C-score	Estimated TM-score	Estimated RMSD
1.83	0.97±0.05	1.8±1.5Å

The confidence of each model is quantitatively measured by C-score that is calculated based on the significance of threading template alignments and the convergence parameters of the structure assembly simulations. C-score is typically in the range of [-5, 2], where a C-score of a higher value signifies a model with a higher confidence and vice-versa. 

TM-score is a metric for assessing the topological similarity of protein structures. TM-score has the value in (0,1], where 1 indicates a perfect match between two structures. Following strict statistics of structures in the PDB, scores below 0.17 correspond to randomly chosen unrelated proteins whereas structures with a score higher than 0.5 assume generally the same fold. 

The root-mean-square deviation of atomic positions (or simply root-mean-square deviation, RMSD) is the measure of the average distance between the atoms (usually the backbone atoms) of superimposed proteins. The RMSD of two aligned structures indicates their divergence from one another.

Stage 2:

Since ovastacin has a eukaryotic origin, this protein undergoes several Post Translational Modifications (PTM) including addition of disulphide bonds and phosphorylation at serine and tyrosine residues.
However, our recombinant protein will be produced in a prokaryotic organism which lacks the necessary genes to carry out PTMs. Hence, we used phosphomimetic variant(s) of ovastacin by changing or mutating the serine to aspartic acid and tyrosine to glutamic acid residues at S99, S156, S244 and Y279 (amino acid numbered with reference to pro-ovastacin) respectively. We then used I-TASSER to predict the structure of this phosphomimetic variant.
The following is the result that we received from the server.

RLLSAASNKWPMGGDGVVEVPFLLSSKYDEPSRQVILEALAEFERSTCIRFVTYQDQRDFISIIPMYGCFSD VGRSGGMQVVSLAPTCLQKGRGIVLHELMHVLGFWHEHTRADRDRYIRVNWNEILPGFEIN FIKSRSSNMLTPYDYSSVMHYGRLAFDRRGLPTITPLWAPSVHIGQRWNLSASDITRVLKLEGCS

C-score	Estimated TM-score	Estimated RMSD
1.84	0.97±0.05	1.8±1.5Å

We used superposition on both the structures of ovastacin to compare them.

Sequence alignment score	RMSD (across all 198 pairs)
936.3	0.197 Å

We went ahead with the sequence of the phosphomimetic variant of ovastacin and determined the ligand binding site using I-TASSER to get a structure with the zinc atom along with the protein further.

This structure showed a consensus with our literature review about the active site of ovastacin. It contains all the conserved features of the astacin family of metalloproteinases, including a deep active-site cleft with three zinc-ligating histidines, an HELMHVLGFWH motif, a Met-turn methionine, and a zinc-proximal tyrosine [2][4].

In the ovum, once this catalytically active zinc ion at the active site is exposed by a process similar to previously mentioned two step activation of crayfish astacin, ovastacin gains its metalloprotease activity and interacts with Zona Pellucida glycoprotein 2 causing cleavage of the ZP2 protein. Hence, we did an initial bioinformatic study of truncated ZP2 protein to check the structure of the recombinant protein.

Zona Pellucida 2 

The human Zona Pellucida (ZP) matrix, a network of thin interconnected filaments, is primarily composed of four glycoproteins, namely, ZP1, ZP2, ZP3, and ZP4. ZP glycoproteins play a critical role in the accomplishment of fertilization by acting as ligands for binding of sperms to the ovum followed by induction of exocytosis of cortical granules to block polyspermy [5]. All four zona proteins share common structural elements including a signal peptide, ZP domain, consensus furin cleavage site (CFCS), transmembrane-like domain, and cytoplasmic tail [6]. The mature protein lacks the transmembrane-like domain after getting cleaved at the CFCS site, It then gets transported to form bonds with other glycoprotein components and form the extracellular matrix [6]. 

The cleavage of mature ZP2 (120 kDa) by ovastacin yields two fragments (90 kDa + 30 kDa) [7] at a conserved cleavage site (167LA↓DE170) [8]. The two protein fragments remain covalently attached via the disulfide bond of ZP2 and mechanistically induce ZP hardening [5].
ZP2 being an eukaryotic protein undergoes several PTMs including disulphide bonds and glycosylations [5]. We planned to express our recombinant ZP2 protein in a prokaryotic system which hasn’t got the required genes to carry out PTMs. However, through integrated human practices with Dr Satish Gupta, we concluded that we will be able to express and utilise recombinant ZP2 without bulk PTMs like glycosylations for the assays. Hence, we predicted the structure using I-TASSER with the sequence of our recombinant protein without any additional modifications.

The following is the result that we received from the server.

The native protein sequence was obtained from ZP2 sequence in uniprot

IDVSQLVNPAFPGTVTCDEREITVEFPSSPGTKKWHASVVDPLGLDMPNCTYILDPEKLTLRATYDNCTRRVHGGHQMT IRVMNNSAALRHGAVMYQFFCPAMQVEETQGLSASTICQKDFMSFSLPRVFSGLADDSKG TKVQMGWSIEVGDGARAKTLTLPEAMKEGFSLLIDNHRMTFHVPFNATGVTHYVQGNSHLYMVSLKLTFISPGQKVIF SSQAICAPDPVTCNATHMTLTIPEFPGKLKSVSFENQNIDVSQLHDNGIDLEATNGMKLHFSKTLLKTKLSEKCLLHQ FYLASLKLTFLLRPETVSMVIYPECLCESPVSIVTGELCTQDGFMDVEVYSYQTQPALDLGTLRVGNSSCQPVFEAQS QGLVRFHIPLNGCGTRYKFEDDKVVYENEIHALWTDFPPSKISRDSEFRMTVKCSYSRNDMLLNINVESLTPPVASVK LGPFTLILQSYPDNSYQQPYGENEYPLVRFLRQPIYMEVRVLNRDDPNIKLVLDDCWATSTMDPDSFPQWNVVVDGCA YDLDNYQTTFHPVGSSVTHPDHYQRFDMKAFAFVSEAHVLSSLVYFHCSALICNRLSPDSPLCSVTCPVSS

We selected the best model according to the C-score and carried out docking on HADDOCK server (High Ambiguity Driven protein-protein DOCKing), a server with a flexible docking approach for the modelling of proteins.

The proteins were docked at the assumed position for binding for the interaction between the active sites; however, the distance between the two binding sites were greater than expected. We could come up with a possible explanation that the protease might be undergoing some conformation changes after activation at the C-terminal region which has got a high number of intrinsically disordered regions and an unknown structure as per the literature [1].

Conclusion:
From the results obtained from the structural models, the structure of recombinant ovastacin and ZP2 remains consistent with the literature regarding the characteristic properties of both the mature proteins. The structures along with the low RMSD values are in accordance with our idea that the chosen active domain of ovastacin and mature sequence of ZP2 remain intact and supports our design for expression of protein which might be functional. Based on the above predictions, we expect that our recombinant proteins might not be affected by structural inflexibility.

Yet these computational predictions act only as supplements for the real practice of expression of the proteins and check for their activity experimentally. However, due to extremely restricted and small window of access to the laboratory our team could not perform a cleavage assay to check the functionality of these recombinant proteins this year. While at the same time our models support our proof of concept of interaction between two stable protein structures as per our predicted structures, we eagerly await the day on which these are shown experimentally.

References:

[1] Körschgen, H., Kuske, M., Karmilin, K., Yiallouros, I., Balbach, M., Floehr, J., Wachten, D., Jahnen-Dechent, W., & Stöcker, W. (2017). Intracellular activation of ovastacin mediates pre-fertilization hardening of the zona pellucida. Molecular human reproduction, 23(9), 607–616. https://doi.org/10.1093/molehr/gax040

[2] Guevara, T., Yiallouros, I., Kappelhoff, R., Bissdorf, S., Stöcker, W., & Gomis-Rüth, F. X. (2010). Proenzyme structure and activation of astacin metallopeptidase. The Journal of biological chemistry, 285(18), 13958–13965. https://doi.org/10.1074/jbc.M109.097436

[3] Roy, A., Kucukural, A. & Zhang, Y. I-TASSER: a unified platform for automated protein structure and function prediction. Nat Protoc 5, 725–738 (2010). https://doi.org/10.1038/nprot.2010.5

[5] Fahrenkamp, E., Algarra, B., & Jovine, L. (2020). Mammalian egg coat modifications and t

[4] Bode, W., Gomis-Rüth, F. X., Huber, R., Zwilling, R., & Stöcker, W. (1992). Structure of astacin and implications for activation of astacins and zinc-ligation of collagenases. Nature, 358(6382), 164–167. https://doi.org/10.1038/358164a0he block to polyspermy. Molecular reproduction and development, 87(3), 326–340. https://doi.org/10.1002/mrd.23320

[6] Gupta, Satish. (2018). The Human Egg's Zona Pellucida. 10.1016/bs.ctdb.2018.01.001. 

[7] Bleil, J. D., Beall, C. F., & Wassarman, P. M. (1981). Mammalian sperm-egg interaction: fertilization of mouse eggs triggers modification of the major zona pellucida glycoprotein, ZP2. Developmental biology, 86(1), 189–197. https://doi.org/10.1016/0012-1606(81)90329-8

[8] Gahlay, G., Gauthier, L., Baibakov, B., Epifano, O., & Dean, J. (2010). Gamete recognition in mice depends on the cleavage status of an egg's zona pellucida protein. Science (New York, N.Y.), 329(5988), 216–219. https://doi.org/10.1126/science.1188178

7.Kill Switch: Reversal of contraception

Overview:

Reversal of contraception can be achieved if all the colonised commensal bacteria can be cleared from the fallopian tube. If there are no GMCs, there wouldn't be artificial production of ovastacin for hardening the ovum and fertilization can take place. In order to achieve this, a kill switch was designed for the bacteria which can be activated whenever the inducer is delivered to the location of the bacteria. The inducer chosen was xylose, this sugar molecule is harmless to the human body since it is ingested through food items too. Also the oviductal fluid composition showed absence of xylose implying that the kill switch shall not get activated unless induced. Next the toxin was chosen for causing bacterial death, along with an antitoxin system in order to prevent leakage of toxin when not needed.

Assumptions:

1. Same amount of xylose concentration reaches all the bacterial cells.
2. Concentrations of toxin greater than antitoxin for a minimum of 5hours induces cell death.
3. XYLR is added to the system first, hence there is continuous production of XYLR first after which toxin and antitoxin systems are added.

Model Development:

Xylose is planned to be delivered exactly like the GMC were delivered using hysteroscope technique. The two questions this model addresses are:
How much concentration of xylose is required for induction of the kill switch?
How much time does it take for the maximum of the population to get killed?
These questions can be answered once a clear picture of protein production is obtained. The genetic circuit was examined and the ODE’s were derived. After the derivation, it was realised that xylose needs to be taken up by the bacteria for the killswitch to work. Hence michaelis menten equation was added explaining Vmax as the maximum rate of xylose uptake, $[S]$ is the concentration of xylose outside the cell which shall be taken as the concentration administered and $K_M$ as the substrate concentration at which reaction rate is half of $V_{max}$. $[S]$ was initially taken as $0.5\%$ of xylose solution which has been shown to induce cell death with the same toxin. Defining

• [xylose]=[X_Y]
• [XYLR]=[X_R]
• [toxin]=[T_Y]
• [antitoxin]=[A_Y]

Equations for the system

1. Xylose Production
   

   \begin{equation}\tag{6.1}\frac{d[X_Y]}{dt}=V_{max}\frac{[S]}{[S]+K_M}-k_{deg}[X_Y]-K_{X}'[X_R][X_Y]\end{equation}

2. XYLR Production
   

   \begin{equation}\tag{6.2}\frac{d[X_R]}{dt}=k_p-k_{deg}*[X_R]-K_X’[X_R][X_Y]\end{equation}

3. Toxin (yqcG) production
   

   \begin{equation}\tag{6.3}\frac{d[T_Y]}{dt}=k_p’\frac{K_D^2}{K_D^2+[X_R]^2}-k_{deg}’[T_Y]-K_Y’[T_Y][A_Y]\end{equation}

4. Antitoxin (yqcF) production
   

   \begin{equation}\tag{6.4}\frac{d[A_Y]}{dt}=k_p’’-k_{deg}’’[A_Y]-K_Y’[T_Y][A_Y]\end{equation}

Calculations and Simulations:

Graphs

1. System prior to xylose induction, with XYLR expressed before toxin and antitoxin cassettes.

   

2. Post xylose induction, with 0.5% of xylose solution injected.

   

Matlab Code

Download

Interpretation:

The above graphs closely resemble the way kill switch should work, but it's better to have toxin levels at zero in the absence of xylose which could be achieved if the Vmax for xylose uptake were higher. According to the current system we answer both the questions as:
The xylose concentration enough to induce cell death is at least 0.0333 M.
The time taken post xylose induction is 10.08 + 5hours = 15.08 ≈ 15 hours.

Xylose Delivery

8.Bio-Safety

Overview

If the bacteria gets flushed out of the body, it shouldn’t survive so that HGT doesn’t occur. This can be achieved if there is a kill switch that gets triggered once it leaves the body. For this a light inducible kill switch was planned based on stress based sigma B activation in general stress response. This whole process consists of a ytva cycle, light induced signalling state to Sigma B activation reaction cascade and sigma B activation to toxin production.

Ytva Cycle: Ytva is continuously produced in the dark state and stays in the dark state until it comes in contact with blue light. It has two domains namely LOV (light oxygen voltage) domain and a C-terminal STAS (sulfate transporter and anti-sigma factor antagonist) domain. LOV domain from these binds to FMN (Flavin Mononucleotide) forming a molecule that has maximum absorbance of 448nm in the dark. Hence, illumination of blue light triggers a photocycle, that includes formation of a signalling state that converts back to dark state with a recovery time of 2600s. Ytva-LOV is considered a monomer and is found to exist as a homodimer.

Reaction cascade: Signalling state of Ytva (S)

Assumptions

1. All the ytva molecules are present in the dark state at t=0.
2. Since the ytva doesn't stay in signalling or excited state for long, its degradation rate is considered only for the dark state.
3. Ytva homodimer binds to the stressosome complex to release eight RsbT molecules.

Model Development

A] ytva activation The reaction rates and states as shown in figure () along with all of the above assumptions and the fact that ytva is produced continuously the following model was prepared. The main focus being ‘S’ i.e. light induced signalling state since it triggers the next reaction cascade.

1. Ytva dark state:

   \begin{equation}\tag{6.1}\frac{d[D_2]}{dt}=k_p-k_{exD}[D_2]+\left(\frac{1}{Q_{yD}}-1\right)k_{pe}[D_2^*]+k_{re}[S_2]-k_{deg}[D_2]\end{equation}

2. Ytva Excited state

   \begin{equation}\tag{6.2}\frac{d[D_2^*]}{dt}=k_{exD}[D_2]+\frac{k_{pe}}{Q_{yD}}[D_2^*]\end{equation}

3. Ytva light induced signalling state
   

   \begin{equation}\tag{6.3}\frac{d[S_2]}{dt}=k_{pe}[D_2^*]-k_{re}[S_2]\end{equation}

The whole reaction cascade can be simplified into the following reaction equations, where, Substrate = Stressosome = [RsbR₄₀RsbS₂₀RsbT₂₀]=[C]
Phosphorylated RsbR and RsbS = [P]
[RsbT] = [T]
Dimer of RsbU = [U₂]
Phosphorylated RsbV = [V’]
RsbV = [V]
RsbW = [W]
Sig B = [B]

[C]+[S₂] $\longrightarrow$ [P]+[S₂]+20[T]
8[T]+8[U₂] $\longrightarrow$ 8[U₂T]
8[U₂]+8[V’] $\longrightarrow$ 8[U₂T]+8[V]
[W₄B₂]₂+8[V] $\longrightarrow$ 4[B]+[W₄V₂]₂

\begin{equation}\tag{6.4}\frac{1}{20} \frac{d[T]}{dt} = V_{max}\frac{[C]}{K_m+[C]} \approx \frac{1}{10} \frac{d[B]}{dt} \end{equation}

V_max can be written as,

\begin{equation}\tag{6.5} V_{max} = k_{tn}[S_2]\end{equation}

Using the above two equations, equation for Sigma B can be written as,

\begin{equation}\tag{6.6}\frac{d[B]}{dt} = 10 k_{tn}[S_2]\frac{[C]}{K_m+[C]}\end{equation}

Toxin (Bovine Dnase) Production: (Bovine Dnase = T_b, Mflon = A_b)

\begin{equation}\tag{6.7}\frac{d[T_b]}{dt}=k_p’\frac{[B]}{K_D+[B]}-k_{dp}’[T_b]-K_D’[T_b][A_b]\end{equation}

Anti-toxin (Mflon) Production:

\begin{equation}\tag{6.8}\frac{d[A_b]}{dt}=k_p’’-k_{dp}’’\frac{[B]}{K_D+[B]}-k_{dp}’[T_b]-K_D’[T_b][A_b]\end{equation}

The work of killswitch is to kill the bacteria when necessary, the toxin used is known to degrade the DNA of the bacterial cell. Since literature studies did not suggest the details regarding time and effect of this toxin, the equation for genome degradation was written as

\begin{equation}\tag{6.9}\frac{\mathrm{d} [G]}{\mathrm{d} t} = -k_{tn_2} [T_b]\frac{[G]}{K_{M_2}+[G]}\end{equation}

Where, [G] represents the genome concentration inside the cell. As bp DNase cleaves the phosphodiester bonds of the DNA, its concentration shall eventually decrease.

Calculation and Simulations

Solving the equations for D, D*, S, B, Tb, Ab we get the following

1. Ytva system

   

   Matlab Code

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   Matlab Code

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2. Toxin-antitoxin system without genome degradation

   

   
   

   

   
   

   

   Matlab Code

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3. bp DNase with genome degradation

   • Initial Genomic concentrations: Assuming that genomic DNA is the only DNA present in the bacteria, and that the cell is plasmid free, the genome concentration is calculated.
   • Taking DNA length = 4220 kbp
   • And Mass of DNA = 6.4 - 6.9 ug for Bacillus Subtilis
   • Moles of DNA = 2.454 fm = 2.454e-15 moles
   • Volume of one bacterial cell = 0.9µm3 = 9e-16L
   • Molarity = 2.72M = 2.72e9 nM
   • Initial Genome concentration = 2.72e9 nM

   

   Matlab Code

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Interpretation

From the above graphs we can say that the ytva activation is an extremely fast process. Which is why an illumination by light leads to rapid increase in Bovine pancreatic DNase in the system. Since within 10secs bp dnase starts taking over, it will start dominating the system eventually resulting in the death of the bacteria. The last graph shows the effect of bp DNAse on the DNA concentration inside the cell, as the concentration reduces as the cell starts dying. It can be concluded that the model shows the expected behaviour of the kill switch once illuminated by light.

References:

Killswitch

1] Bacterial Cells Experiencing Sudden Chromosome Loss Elbaz and Ben-Yehuda Maya Elbaz, Sigal Ben-Yehuda, Petra Levin and R. John Collier. Doi: 10.1128/mBio.00092-15

2] Following the Fate of Bacterial Cells Experiencing Sudden Chromosome Loss Maya Elbaz and Sigal Ben-Yehuda and Petra Levin and R. John Collier,mBio,Doi: 10.1128/mBio.00092-15

Biosafety

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