Team:IISER Tirupati/Model

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Experiments
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Why?

While designing our project, we had many questions regarding the behaviour of CoCa coli within a biological system. We needed concrete answers regarding its journey through the digestive tract, and a guarantee that it has high chances of encountering the tumour on the wall of the colon. As none of these questions could be approached experimentally, we decided to mathematically model them.

Additionally, we constructed various combinations of lldP, lldR and lldD, each one resulting in IL12 production at a particular lactate concentrations. To fine-tune this process, we also had combinations of strong, medium and weak promoters and RBS. Rather than blindly characteristing all these constructs, we modeled the "lactate" operon, and brought down the number to a few useful constructs, and understood our bacteria functioning much better.


What?

Module-1: The Lactate Operon

As mentioned above, we wanted to understand the working of the various lldP, lldR and lldD; strong/medium/weak RBS & promoter combinations - to choose for the best construct in a given tumour scenario. Particularly, we were interested in knowing -

  • Can a particular construct give a “spike” in IL-12 production level at a threshold concentration of lactate? Will the IL-12 production level remain negligible below this lactate threshold?
  • What is this threshold lactate concentration for each construct?
These questions are answered in “The Lactate Operon” module!

Module-2: CoCa coli in the Colon

What happens when a colon cancer patient ingests CoCa coli?
We propose on using a capsule containing CoCa coli to administer the probiotic to patients. This capsule is a type of CDDS (colon-specific drug delivery systems) which uses either the unique pH, microflora or enzymes of the colon to dissolve and release its contents - which shall be CoCa coli [1]. The release of CoCa coli can be controlled to occur specifically during the transition from the cecum to the ascending part of the large intestine. This ensures that CoCa coli has no chance of encountering the rest of the body - all it ever sees is the colon.

Once released in the large intestine, what are the chances that the bacteria reaches the microenvironment of the tumour?
This question is explored in our “CoCa coli in the Colon” module, which studies the fluid dynamics of bacteria in the colon as a first step to understanding CoCa coli’s behaviour after being released in the cecum.

Module-3: CoCa coli vs Cancer

Given a particular stage of colon cancer, what are the chances that CoCa coli can effectively reduce its size?
After reaching the tumour microenvironment, we first have CoCa coli’s Recognition Module, that ensures the secretion of IL12 due to the presence of high lactate and the RPMrel binding site (specifically only in this region, and not anywhere else in its journey through the alimentary canal!) and now, the Secretion Module comes into the picture. To answer the above question, we have the “CoCa coli vs Cancer” module that predicts the type (or construct!) and the number of bacteria that needs to be administered to theoretically curb tumour growth, and reduce its population size to zero.



Results

The Lactate Operon
Our model proposes a set of equations based on the lactose operon, which mathematically describes the working of the genomic and exogenous lactate operon. Due to the scarcity of literature and experimental data, we haven’t solved these equations but instead, left them for further study by other teams.

CoCa coli in the Colon
Our major findings are -

  • Usage of a motile bacteria is a must to target tumours within the colon to overcome the velocity of colonic fluids.
  • We obtained an expression to find out the final concentration of bacteria that is attached to the tumour. This is useful to calculate the initial number of bacteria that needs to be administered via the capsule!
    • CoCa coli should ideally be administered before food intake to minimise detachment due to shear forces.

CoCa coli vs Cancer
We obtained simulations for tumour growth with and without IL-12 administration. It was observed that with a reported literature value for the constant production rate of IL-12 (shown to experimentally degrade tumour), both high and low detectable initial tumour populations could be reduced to zero -hence validating our use of IL-12.

The Aim!

To understand the dynamics of Il-12/sfGFP production under combinations of lldP, lldR, lldD, strong, medium, weak promoter and RBS.

We particularly hoped to answer:

  • Can a particular construct give a “spike” in IL-12 production level at a threshold concentration of lactate? Will the IL-12 production level remain negligible below this lactate threshold?
  • What is this threshold lactate concentration for each construct? Check out the various constructs from our project Design.

    Given the complex interactions between these various systems, we decided to study the well-known lactose operon, and adapt this approach for our problem.

    Step 1:

    The first step is to account for the population of cells, and write down the equations for change in concentrations of RBS responsible for the translation of lldP, lldR and lldD mRNAs.

    This is represented in the following manner for lldP,R or D, represented as x



    Where,
    MP = concentration of RBSs of mRNA for lldP
    D = concentration of lactate promoters
    PτP”(t)(OP) = probability of RNA polymerase complex being bound to the promoter region, and is ready to initiate transcription; including a delay term which factors in delay between transcription and translation

    The equations for lldR and lldD have a similar format (whose RBS concentration is expressed as MR and MD).

    Now, we need to determine the value for each of the terms in the above equations. These are found and estimated in a stepwise manner.

    Step 2:

    Description of the system through equilibrium reactions



    Step 3: Determination of P(OP)

    The net probability that the polymerase-complex will bind to the operator is given by the sum of probabilities that O1 is free, O1 is bound to both the repressor and to lactate, and O2 is free. Using this, we get the following equation -



    The above terms represent -



    where the rate constants are as described from the above cell-system equations.

    Step 4:
    Equations for the translation of mRNA



    Terms in the above reaction represent -
    kP = maximum translation rate by the RBS of lldP
    τP” = time necessary to translation of a polypeptide chain of lldP
    𝛾P = lldP protein degradation rate
    μ = average growth rate of bacteria
    e -μ𝜏P = dilution of mRNA concentration due to cell growth in the time interval [ 𝜏 - 𝜏P, t]
    M𝜏P(t) = mRNA of lldP concentration
    (similar terms are used for lldR, lldD and for sfGFP/IL-12)

    Step 5:
    Accounting for the intracellular lactate concentration





    The terms in the above equations represent -
    Le = concentration of extracellular lactate
    KLBLQ = gain of intracellular lactate obtained due to Q (which refers to the protein of lldP)
    klBl(L) Q = loss of intracellular lactate to extracellular fluid, because of the reversible nature of permease
    𝜙MM(L)B = this term accounts for the lldD (dehydrogenase) mediated loss of lactate due to conversion of lactate to pyruvate
    (𝛾L+𝜇)L = this term accounts for the natural degradation and dilution of lactate
    𝜅LB = this term accounts for the loss of lactate due to binding of the regulator

    Following the construction of these equations, we need to vary the extracellular lactate concentrations (LE) and predict the changes observed in IL-12/sfGFP production. Additionally, the transcription and translation rates fit into the equations would vary upon the strength of the RBS and promoters used.

    However, the parameter values required for solving the above system of linear equations can only be obtained after extensive study of the lactate operon - which has neither been performed in literature, nor could be experimentally completed by us. We end this module by proposing the above set of constructed equations to be a mathematical equivalent of the lactate operon - with variations in parameters giving accurate descriptions of the genomic and exogenous systems.

    Reference: Mackey et al., 2014; The utility of simple mathematical models in understanding gene regualtory dynamics

The Aim!

To study the flow of bacteria from its point of release in the cecum until the tumour-site; by understanding the flow of bacteria until this point, and its attachment kinetics.

How?

Background

On searching up the work of others in this topic, we found a previous iGEM project (belonging to Team: Greece, 2017, [2]), whose work in modelling provided us with a lot of relevant answers. Based on a fluid dynamic model for cancers in the colon, and a correlation between the shear rate of fluid flow and tumour surface attachment, they concluded that a considerable number of bacteria can attach to the tumour.

However, this study focused on tumours in the early/ascending parts of the colon - without taking peristalsis into consideration. We looked into the literature on this and found that the human colon has shear rates predicted at values beyond the low-shear region limits [3]. A low-shear value is required to ensure that the bacterial number doesn’t get depleted due to trapping of bacteria in the high-shear regions [4]. This poses a problem to our system, as CoCa coli may get trapped in these high-shear regions, and not reach the tumour present ahead in the colon.

Based on what we learnt from these papers, we decided to take a fresh approach to this problem. A rigorous model of our work has been described below.

Our Approach

Basic assumptions

  1. The fluid inside the colon is considered to be Newtonian and incompressible
    This allows us to describe the fluid with the following equations, which are respectively the incompressibility equation and the Navier-Stokes equation.





    ( V is the velocity of the fluid, p is the pressure, 𝜂 is the viscosity of the fluid )

  2. The Reynolds number for bacteria flow inside the colon is low
    This can be derived via the formula for Reynolds number, Re; which represents the ratio of inertial to viscous forces.



    (Reis the Reynolds number, 𝜌 is the density of the fluid, a is the linear size of the immersed object u is the velocity of the immersed object, 𝜂 is the viscosity of the fluid ) Substituting the values for the above terms for the movement of Escherichia coli in water, (𝜂 = 10-3 Pa.s; 𝜌 = 103 kg m-3; a = 10-6 m; and say the velocity of the bacteria, u = 10-5m/s) we get Re= 10-5 << 1. This validates the above assumption, given that colonic fluids are more viscous than water.

    Under the above two assumptions, the terms on the LHS of Navier-Stokes equation gives zero; leading us to the Stokes equation, described as follows -

    This, along with the incompressibility equation need to be solved simultaneously, under specific boundary conditions to arrive at the solution for our system.

  3. Assumptions about the colon -
    • The colon is assumed to be a cylinder (or tube) of radius R
    • The “no-slip” boundary condition is taken to be valid at the inner walls. This means that the fluid layer in contact with the walls of the cylinder is at rest; or tangential velocity of fluid at the walls is zero (this was based on literature survey that gave us proof that despite invalidation of “no-slip” in colonic physiological scenarios, our specific picture can fall into “no-slip” [5,6]).
    • A pressure gradient is assumed to exist along the length of the tube; validated by the existence of pressure difference along the length of the colon.




    Following solution of the Stokes equation, along with the no-slip boundary condition of the walls of the “tube”; we arrive at the Parabolic or Poiseuille flow system; under which the equation for fluid velocity is given as follows -



    ( V is the velocity of the object, Vo is the initial velocity of the object, R is the radius of the colon )

Bacterial Movement

As the final aim of our project is to understand the concentration of CoCa coli along the length of the colon, and understand its flow behaviour with respect to the tumour; we need to study the flow of a large number of bacteria released simultaneously from the capsule. This original concentration, however, will vary with the colon position under study, and the time after bacterial-release from the capsule.

Let c(r,t) be the concentration of CoCa coli. Then the following equation describes the change in the concentration of a bacteria with time; considering terms that account for the diffusion of the bacteria; and the z-component of colonic fluid velocity.



(c is the concentration of bacteria at a given point and time, Dtis the translational diffusion constant of E. coli, vz is the z-component of velocity) In the above equation, the calculation of Dtvaries with other parameters, involving the specifications of the bacteria.

  1. Non-motile E. coli
    When the non-motile system is considered, the translational and rotational components of its motion (or velocity) are decoupled. This can be understood from the following -



    ( Dtis the translational diffusion constant, Dr is the rotational diffusion constant, kBis Boltzmann’s constant, T is the temperature ∼ 310K, a is the effective hydrodynamic radius of E. coli ∼ 10-6 m, 𝜂 is the viscosity of the fluid ∼ 10-3 Pa.s; note that the equations above are valid for the radius of a spherical particle, due to the rod-shaped nature of E. coli, we instead take the effective hydrodynamic radius ) This gives the approximate values for the above terms as Dt= 2 x 10-3 m2s-1 and Dr = 0.17 s-1. As can be seen, Dr << D; which results in the decoupled scenario. (However, this is true only in dilute solutions, that is, when the concentration, c is low!)

  2. Motile E. coli
    Here, rotational diffusion is strongly coupled with translation. As a result, we need to consider the effective diffusion rate following coupling, instead of Dt. This is given as -



    ( Deff is the effective diffusion constant for a motile bacteria, due to coupled translational and rotational motion, u is the velocity of E. coli ∼ 10-5 ms-1 ) Furthermore, DrcH can be much larger than the thermal value, Dr. Hence, the above is re-expressed into -



    ( 𝜏t is the time beyond which the bacterium loses all memory of its initial orientation, and is ∼ 10s ) Whose value comes out to be approximately equal to 3 x 10-9 m2s-1.

  3. Motile E. coli when peristalsis is occurring in the colon
    From literature, this value DPeff is approximately equal to 10-6 m2s-1 [7]. This needs to be considered in a separate scenario given the high variation in the effective diffusion in these two conditions. Coming back to the equation for the change in bacterial concentration with respect to time, we observe that it accounts only for the z-component of colonic fluid flow.

Why is this?

Under the influence of the fluid flow, bacterial flow also gets affected. However, a consideration of the drift-vs-diffusion study along the rest of the directions lets us safely ignore the other components of velocity.

Let x be the distance travelled in time ‘t’
For translation due to diffusion along the direction of ‘x’, we have -



For the drift of bacteria due to velocity of fluid flow (along the x-direction only); called advection -



Comparison of the above two values gives us -



Substitution of the values leads to the LHS term in the above equation to be approximately 10-4 << 1. Hence, diffusion in the direction of external velocity component (with velocity, V) can be neglected in the presence of V.

Effective Target Length of the Colon

Let us consider the release of bacteria at (xo, yo) at t=0. After time ‘t’, the bacteria “cloud” will diffuse across the tube width to reach a point (x, y). This is described, after taking diffusion into account as -



If we consider (xo, yo) to be equal to (0,0) for simplicity, we get



Now, let tibe the time required for the bacteria to reach the wall. This implies that the position of the bacteria at time, tigiven by (xi yi) must be -



On substitution, we get -





Hence, we have determined the time necessary for the bacteria to reach the walls (considering the movement of bacteria along the x-y directions only). However, during this time, the bacteria shall along move along the z-direction due to the colonic-fluid velocity. This can be better described as the drift of bacteria along the z-direction; through a distance as zi.

Calculation of zi is performed by averaging the distance travelled until ti.





Substitution for ti from the above obtained derivation leads us to the final expression -



Calculation of zi under the above-described conditions for variant D is (velocity values from [8,9]):

Table 1
Condition Value of zi
1. Non-motile E. coli 2 x 103 m
2. Motile E. coli 0.16 m
3. Motile E. coli with peristalsis occurring in the colon 0.11 m


What does this tell us?

Based on the above definitions, zi represents the distance travelled by bacteria in the time that it takes to reach the wall of the colon. This implies that for z < zi the bacteria does not reach the walls of the colon. As tumour growth occurs along the walls, this parameter is crucial in describing the cancers which can be targeted via CoCa coli.

When non-motile bacteria are considered, zi has a value in kilometres, which is highly unfavourable (it’s more than the length of the colon! ). Hence, we propose the use of motile bacteria for our purpose.

But, the release of bacteria need not be at the centre!

A simulation study gives us a picture of bacterial transport following released along any cross-sectional location of the colon. However, a simplistic approach along 1D gave us the necessary answers.

Consider the diffusion only along the x-direction (1D diffusion). The value of zi under such a picture turns out to be -



Which results in a smaller value of v - describing an efficient targeting for tumour closer to the cecum!

For an absorbing boundary condition (which can be inferred from the no-slip assumption), we perform the following analysis. Let ‘x’ be the point where the bacteria is released and let W(x) represent the time taken to reach the wall. Then,



With W (x = R) = 0 and W (x = -R) = 0

The solution for the above equation is of the form -



Then, the mean time taken to capture the bacteria released at ‘x’ anywhere between x = R and x = -R (taking the averaging allows to predict the system for a randomised delivery system, where bacterial release can occur at any ‘x’! )



The corresponding value for zi is given by -



This value is of the same order of magnitude and slightly smaller that the zi that we calculated earlier!

The value described her is a better approximation of the exact time and location at which the bacteria reaches the wall of the tumour; given randomised bacteria release. Interestingly, zi values calculated on the basis of previous assumptions (ref. Table 1) represent another important quantity. Read on!

Diffusion across the Width of the Colon

The diffusion of the bacteria along the x-y component is given by the equation (as explained earlier) with vz = 0 [10]-



If we consider that at t = 0, all the bacteria are concentrated at the point (0,0) along (x,y) and are “released” together by simultaneous diffusion; the description of this system reveals the following - Take -



The above described situation can be written as -



( cois the initial concentration of bacteria released from the capsule, 𝛿 is an delta-function taking values of when its input corresponds to 0 ) Then given the initial conditions, the following equation describes the collection of bacteria to exhibit a Gaussian spread - a Gaussian waveform, that increases in width with time.

We need to find out the maximum concentration, c at (x,y) = R; hence, description of this leads us to -



Whose solution gives us



Hence, we can conclude that c(R,t) or the bacterial concentration on the walls of the colon at a given time ‘t’ is maximum when the previously described parameter, tm. Hence, what we originally described to be the time required for the bacteria to reach the wall when released at the centre, is actually the time taken by bacteria to attain maximum concentration on a particular point on the wall (at the ‘z’ corresponding to tm ).

Also,



Which describes the concentration attained at (R, zi, tm).

What does this tell us?
The above equation is extremely useful, as it allows us to predict the initial concentration of bacteria that needs to be supplied via the capsule (given by co) to achieve a particular, desired concentration of bacteria in the tumour microenvironment.

But we do not have unlimited bacteria supply!

As we progress through the z-axis in the colon, we see that diffusion ensures the movement of bacteria towards the walls of the colon. This means that the bacterial concentration will reduce as we go forward [11]. Hence, there must exist a threshold value of ‘z’ beyond which the concentration of bacteria is too little to be of any use to us.

This distance is described as zf which is reached at tf. Say (n is taken to be an arbitrary constant, whose value is determined via the experimental/modelled value for cth),



Then,





We define cthreshold = cth as the threshold concentration below which cell attachment is not sufficient to be of biological use (refer below for our attachment studies!). The,





In the time tf, the distance covered along the tube = Votf ; and zf is the distance along the wall beyond which c (R,t) < cth. Assuming the shape of the wavefront to remain relatively constant along the length of the tube, we get -
or




Then, calculations for zf leads us to -





Hence, zf = 2n(zi)

What does this tell us?
The obtained values for zi and zf allows us to determine the target length within the colon; defined as the region within the colon, along which the tumour along the walls can be effectively targeted through administration of CoCa coli.

Attachment-Detachment Kinetics

We conclude from the above module that the bacteria can reach the walls on the colon; specifically the region where the tumour is present. Following this, attachment of the fimbriae on the bacteria to the 𝛼5𝛽1 integrin occurs due to the presence of RPMrel. But, the fluid flow in the colon may lead to detachment of this bacteria - how can this be studied?

Assuming the no-slip condition on the walls of we see that once CoCa coli reaches the tumour surface, it stays there. However, bacteria have a finite size (≈10-6 m), and hence, could be affected by the tangential velocity components. Additionally, we need to consider the effect of shear forces. Some references actually predict (experimentally/theoretically) the enhancement of attachment due to shear forces [12], however, this is dependent on the type of bonds formed.

We took a generalised approach, and instead studied the effect of shear forces on the detachment kinetics. Based on literature survey, we considered a thermodynamic model for the attachment of multiple ligands onto one cell [13]. The relationship between the detachment rate and shear velocity has been observed to follow different patterns in different ranges of shear velocity. For simplicity, we consider only the high shear velocity range, and show that attachment occurs to a reasonable extent even under this extreme condition.

Let F be the shear force acting on the top surface of the cell, l2 be the area of the bacteria parallel to the direction of flow, 𝜂 the viscosity, and 𝛾. the shear velocity. Then,



F exerts a torque of bF/2 along the direction of z about the center of the cell. Once the end point O, facing towards the flow direction touches the surface, the torque tries to rotate the cell about O. Since the bacterial cell is rigid, this results in stretching of the bonds until the point O.



Let So be the maximum length of an adhesive bond, beyond which the bond breaks. The front of the bonded region (as shaded in the figure), situated at a distance 'r' from O moves towards O at a rate of say, dr/dt. Then,



Here, Fθl is the clockwise torque about O due to shear, ksor represents the anti-clockwise torque about O due to the extension of bonds (with an effective spring constant of k).

Under conditions of strong shear; consider Rd to be the inverse of 𝜏R (and 𝜏R to be the time constant of cell detachment) -



We also use the following, regarding the proportion of attached bacteria -









In practice, A⍺ ≠ 0. This means that even after an infinite amount of time, some amount of bacteria remain attached to the surface - which is good for us. But, for simplicity, we consider A⍺= 0. Then, in the above equation, we get



(Where N(t) is the number of cells attached to the surface after time, t; Re is the radius of the colon, Vo is the maximum observed velocity of fluid in the colon and Ao is the initial concentration of bacteria attached to the surface, which we approximate to be the same as the concentration of bacteria that reach the surface) From the previous sections, we get -



For the three scenarios described above (non-motile bacteria, motile bacteria and motile bacteria in a fluid undergoing peristalsis), we substitute for the values of maximum observed velocity and arrive at the following values for the time constant of detachment (𝜏R)-


Table 2
Conditions Value of 𝜏R
1. Motile E. coli (under conditions of no peristalsis) 1.17 x 104 s m
2. Motile E. coli (with peristalsis occurring in the colon) 58.8 s


What does this tell us?

𝜏R represents the time constant for cell detachment, and its inverse R represents the rate of detachment. Hence, higher values of 𝜏R are more favourable for bacterial attachment; and we propose administration of CoCa coli between food consumption (to ensure minimum peristalsis). These values can further be used to calculate the number of bacteria that remain attached after a time, t; given the initial number of bacteria that are administered through the capsule.





References:

  1. Ref.: Amidon et al., 2015; Colon-Targeted Oral Drug Delivery Systems: Design Trends and Approaches, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4508299/
  2. iGEM Team: Greece, 2017; https://2017.igem.org/Team:Greece
  3. Takahashi, 2010; Flow Behaviour of Digesta and the Absorption of Nutrients in the Gastrointestine, https://www.jstage.jst.go.jp/article/jnsv/57/4/57_4_265/_pdf
  4. Rusconi et al.,2014; Bacterial transport suppressed by fluid shear, https://www.nature.com/articles/nphys2883?page=1
  5. Tripathi et al., 2011; Stokes flow of micro-polar fluids by peristaltic pumping through tube with slip boundary condition
  6. Chaube et al., 2015; Peristaltic Creeping Flow of Polar Law Physiological Fluids through a Non-uniform Channel with Slip Effect
  7. Mackey et al., 2014; The utility of simple mathematical models in understanding gene regualtory dynamics
  8. Cremer et al., 2017; Effect of Water Flow and Chemical Environment on Microbiota Growth and Composition in the Human Colon
  9. Uno, 2018; Colonic Transit Time and Pressure based on Bernoulli’s principle
  10. Dickinson & Cooper, 1995; Analysis of Shear-Dependent Bacterial Adhesion Kinetics to Biomaterial Surfaces
  11. Rusconi et al., 2014; Bacterial transport suppressed by fluid shear
  12. Thomas et al., 2002; Bacterial adhesion to target cells enhanced by shear force
  13. Jotten et al., 2019; Correlation of in vitro cell adhesion, local shear flow and cell density

Why?

Following the long journey of CoCa coli through the alimentary canal and the colon, we have shown in the previous module that our bacteria can indeed reach the tumour microenvironment. The lactate concentrations in this region will result in the production of IL-12 - which then has the potential to recruit various cells of the immune system, and mounts an attack on the tumour.

This vague understanding, however, does not give us any details on the amount of IL-12 production needed to use CoCa coli as a therapeutic agent. We need to the exact concentration that would be required to bring down the tumour cell population to zero - this is what this module hopes to achieve.

The Aim!

To study the population dynamics of tumour after administration of Coca coli.

How?

The model presented below is an extremely simplified form of the interactions that occur in our body, but we hope that it shall be sufficient in calculating parameters necessary for CoCa coli.

Literature survey revealed that, a few key players are involved in attacking the tumour cells following activation by IL-12. These are the Natural Killer (NK) and Cytotoxic (or CD8+T) cells (whose effect on the tumour is similar to any other pathogen-based immune response). Additionally, cytotoxic T cells cannot function of their own in response to an external antigen - they instead require antigen presenting cells (APCs), that present a modified form of the antigen to T cells - which can then, mount an immune response. We have considered one of the major APCs in the human body - namely, dendritic cells.

Following this, we constructed a system of nonlinear ordinary differential equations, each one of which describe the population change of an essential cell type/ for IL-12 [1,2]. One important point to note here is that IL-12 causes an effect on the tumour via the NK cells and CD8+T cells, and not directly.

Our Equations!

The terminology -
T(t)=tumour cells
L(t)=CD8+T cells
N(t)=NK cells
D(t)=Dendritic cells
I(t)=amount of IL12 administered
All other variables are parameters, whose values are mentioned in the end.

Eqn1. - For tumour cells



This terms the change in tumour population with time. It takes into account the logistic growth of the tumour cells, their death due to the NK cells and due to the CD8+T cells.

The terms -

  1. Logistic growth rate for the tumour, which can be safely considered due as we are trying to study the tumour population only in s localised region; and due to the Allee effect, which suggests that the birth rate of tumor cells increases with cell number in the regime of small population size [3].
  2. The effect of NK cells, are assumed to be proportional to the product of effector and tumour cell population (a simplified version of a predator-prey model); where the tumour cell acts are the prey and the NK cells as the predator [4]
  3. The effect of CD8+T cells was experimentally modelled and shown to follow a complex dynamics - the tumour decrease rate depends on the fraction of CD8+T cells to tumour cells, and has a saturation effect [5].


Eqn2. - For CD8+ T cells



The population growth rate of CD8+T cells in the above equation takes into account their production and degradation/inactivation rates.

The terms -
  1. The first three terms are the death rate, inactivation due to tumour and self-inactivation rates respectively each of whose format is justified by literature review [6,7]
  2. Activation of CD8+T cells can be caused due to -
    Indirect - The debris of tumour produced due to the action of NK cells, can lead to antigen presentation by dendritic cells to the CD8+T cells
    Direct - The tumour itself can directly cause dendritic cells to present antigens to the CD8+ T cells. However, this terminology is slightly controversial, and we investigated our system with and without this term to finally include it in our analyses [8]
  3. Additionally, the effect of IL-12 on the activation of CD8+T cells is considered through a Michaelis-Menton equation due to the ligand (IL-12)-receptor(on the surface of CD8+ T cells) interactions [9]


Eqn3. - for NK cells



Similar to the above, the production and degradation/inactivation rates of NK cells are accounted for.
The terms -
  1. NK cells are present in our system even before the presence of the tumour, and hence have a constant production rate (independent of other tumour-related factors) [10]
  2. The next terms indicate the natural death rate and an inactivation rate due to the tumour.
  3. The effect of IL-12, again, is taken through a Michaelis-Menten equation [9]


Eqn4. - for dendritic cells



The terms -
  1. The dendritic cell population has a constant production rate and self-dependent death rate, independent of the presence of the tumour [11].
  2. Additionally, its maturation is affected by the interaction with a presentable antigen; this effect is accounted for by the final term, which considers this effect to be proportional to the number of CD8+T cells and also reaches a saturation due to the same [11].


Eqn5. - for IL12

We have two approaches for the Il-12 part of our system (all the equations given below are proposed specific to our mode, and have not been taken from literature!)
  1. Using the above four equations, the amount of IL-12 required to reach a steady state tumour population level (this steady state needs to be lower than the initial tumour population level!) can be calculated theoretically.
  2. The production of IL-12 in the system can be accounted for by an equation, which can then be used to analyse our system. The equation has been taken into account by this form -





    In the above, the production of Il-12 by our bacteria is accounted for by a heaviside step function (which takes values of 1 for a positive input and 0 for negative input values). This input value is calculated through the difference between L(t) and c(t), where L(t) is the lactate concentration in the tumour microenvironment, and c(t) is the discrete threshold lactate value recognised by our bacteria (CoCa coli can produce IL-12 only after it detects a threshold concentration of lactate!). IL-12 also has a half-life, which is included through a natural death rate term. In the next equation, f(t) represents the lactate concentration in the tumour microenvironment, f(to) represents the lactate concentration at the initial time point (when CoCa coli is administered). Additionally, the contribution by the tumour population size is controlled by the factor 'p'. If the tumour size increases, 'p' shall be negative; and if the tumour size decreases 'p' shall be positive. This accounts for the necessary increase and decreases in lactate concentration, with increase and decrease in tumour size.


One drawback in this equation - IL-12 can also be produced due to the immune system’s natural response against tumour, and due to recognition of the bacterial cell-wall (especially, LPS). This phenomenon has not been studied well enough to be modelled, and hence, has been omitted in our equations. . However, this can cause changes in the IL-12 concentrations, especially given the probiotic nature of our immunotherapeutic technique.

The next step was to determine the parameter values for all the above equations specific to colon cancer scenarios and determination of the initial conditions for simultaneously solving the differential equations.

The parameter values are taken from well-established tumour models, which monitor tumor-immune interactions. Note that our references for the parameter values may themselves have sometimes estimated the value of the parameters.



The above equations were solved simultaneously, after being made specific to colon cancer, due to the parameter values used. These are solved using the Adam_Bashforth 5th order numerical integration method with step size 10-6. All the codes were written in Python.

These were studied both with and without additional IL-12 administration. Constant degradation rates were considered in it.

The values for λ were both attempted through educated guess-work and taken from [12]. This represents the maximum IL-12 production rates.

Results obtained -



  1. Without IL-12 administration :

    (1) Non-detectable tumour

    Initial conditions: Tumour(0) = 1e+05, CD8+T(0) = 3e+02, Dendritic(0) = 3e+04, NK(0) = 3e+04, a=0.002

    (2) Detectable tumour

    Initial conditions: Tumour(0) = 1e+06, CD8+T(0) = 3e+02, Dendritic(0) = 3e+04, NK(0) = 3e+04 a=0.002

  2. With IL-12 administration :

    (1) Non-Detectable Tumour Size
    Our modelling shows that even without the effect of IL-12, non-detectable tumour population can go to zero due to the effect of the naturally present immune system

    (2) Detectable Tumour Size (at low population number) with two tumour growth rates

    Initial conditions: Tumour(0) = 1e+06, CD8+T(0) = 3e+02, Dendritic(0) = 3e+04, NK(0) = 3e+04 IL-12 production rate (λ)= 3.3e+10 IU/L/day, Tumour Growth Rate(a) = 0.02



    Initial conditions: Tumour(0) = 1e+06, CD8+T(0) = 3e+02, Dendritic(0) = 3e+04, NK(0) = 3e+04 IL-12 production rate (λ) = 3.3e+10 IU/L/day, Tumour Growth Rate(a) = 0.002

    (3) Detectable Tumour Size (at high population number)

    Initial conditions: Tumour(0) = 3e+06, CD8+T(0) = 3e+02, Dendritic(0) = 3e+04, NK(0) = 3e+04 IL-12 = ( λ = 3.3e+10 IU/L/day), a=0.02
    (Please note that, although due to the limited computational power, we could run the simulations only till day 50, we observe a clear decrease in the tumour cells indicating their approach towards zero in due time)



    Initial conditions: Tumour(0) = 3e+06, CD8+T(0) = 3e+02, Dendritic(0) = 3e+04, NK(0) = 3e+04 IL-12 = ( λ = 3.3e+10 IU/L/day), a=0.002

The results obtained show a clear death of the tumour cells after administration of IL-12 in contrast to the situation where detectable tumour showed growth without administration of IL-12.





References:

  1. Xeufang et al., 2015; A Mathematical Model of Tumor-Immune Interactions Incorporated with Danger Model; Mamat et al., 2013
  2. Mamat et al., 2013; Mathematical Model of Cancer Treatments using Immunotherapy, Chemotherapy and Biochemotherapy
  3. Johnson et al., 2019; Cancer cell population growth kinetics at low densities deviate from the exponential growth model and suggest an Allee effect
  4. Kaur & Ahmad, 2014; On Study of Immune Response to Tumor Cells in Prey-Predator System
  5. de Pillis et al., 2005; A Validated Mathematical Model of Cell Mediated Immune Response to Tumour Growth
  6. de Pillis et al., 2005; A validated mathematical model of cell-mediated immune response to tumor growth
  7. Rosenberg SA, Lotze MT, Cancer immunotherapy using interleukin-2 and interleukin-2-activated lymphocytes, Annu Rev Immunol 4:681–709, 1986)
  8. Diefenbach et al., 2001; Rae1 and H60 ligands of theNKG2D receptor stimulate tumor immunity
  9. Kirschner & Panetta, 1998; Modeling immunotherapy of the tumor-immune interaction
  10. Kuznetsov et al., 1994; Nonlinear dynamics of immuno-genic tumors: parameter estimation and global bifurcation analysis
  11. Castiglione & Piccoli, 2006; Optimal control in a model of dendritic cell transfection cancer immunotherapy
  12. Liu et al., 2017; Modified Nanoparticle mediated IL-12 Immunogene Therapy for Colon Cancer
  13. Robertson, 1999; Immunological Effects of Interleukin 12 Administered by Bolus Intravenous Injection to Patients with Cancer
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