### Hans H. Diebner's Research

## Polyclonal T Cell Niche Model

dx

k

r

i = 1, ... , M (#species = 2 #clones)

j = 1, ... , N (#niches)

_{i}/dt = 1/τ_{i}x_{i}(1 - x_{i}/k_{i})k

_{i}= v_{i}Σ_{j}r_{ij}r

_{ij}= a_{ij}x_{i}p_{j}/Σ_{k}a_{kj}x_{k}i = 1, ... , M (#species = 2 #clones)

j = 1, ... , N (#niches)

Number of clones:

Each clone has a healthy and an oncogenetic variant, i.e.,

Number of species:

Number of niches: *M/2 = 20*Each clone has a healthy and an oncogenetic variant, i.e.,

Number of species:

*M = 40**N = 20*

healthy clones

*i = 0,...,M/2*

preleukemic clones

*i = M/2+1,...,M*

Initial values:

*x*

∀ i = 1,...,M/2

_{i}(0) = Rand[1.5,2.5]∀ i = 1,...,M/2

*x*

∀ i = M/2+1,...,M

_{i}(0) = Rand[0.5,1.5]∀ i = M/2+1,...,M

cell cycle time:

*τ*

_{i}= 8.0 ∀ i = 1,...,M/2*τ*

_{i}= tuneable ∀ i = M/2+1,...,MNiche sizes:

*p*

_{j}= 100 ∀ i = 1,...,NResource efficiency:

*v*

_{i}= 1 ∀ i = 1,...,M/2*v*

_{i}= tuneable ∀ i = M/2+1,...,MAvidity:

*a*

+ Gaußian(u,βu) if k = j

_{kj}= Gaußian(s,βs)+ Gaußian(u,βu) if k = j

*a*

_{kj}= Gaußian(u,βu) if k ≠ j*k=i if i≤M/2; k=i-M/2 if i>M/2*

*s = 1 ∀ i = 1,...,M/2*

*s = tuneable ∀ i = M/2+1,...,M*

*u = 1/N ∀ i = 1,...,M/2*

*u = tuneable ∀ i = M/2+1,...,M*

*β = tuneable*

References: Sebastian Gerdes, Sebastian Newrzela, Ingmar Glauche, Dorothee von Laer, Martin-Leo Hansmann and Ingo Roeder Mathematical modeling of oncogenesis control in mature T-cell populations, Front. Immunol., 21 November 2013 | doi: 10.3389/fimmu.2013.00380.

Hans H. Diebner, Jörg Kirberg, and Ingo Roeder: An evolutionary stability perspective on oncogenesis control in mature T-cell populations. J. of Theor. Biol., 389, 88-100, 2016.

The simulation shows time series of the growth dynamics of 20 T cell clones over 1000 time unities. A T cell clone is uniquely defined by its T cell receptor (TCR). Each clone has a preleukemic variant, giving rise to a number of 40 species in total. The 20 healthy species' concentrations are green-coloured whereas the remaining concentrations of the 20 oncogenetic variants are red-coloured. Both subpopulations are independently parametrised.

It is believed that the total T cell population always contains a few preleukemic mutants which are characterised by different growth or proliferation parameters, respectively. It is the aim of the model to show that in a polyclonal situation the immune system is able to suppress the preleukemic variants, whereas in monoclonal arrangements the oncogenetic clone variant is able to survive and dominate over the healthy variants of the clone. The default parameter values of the simulation are chosen such that the healthy variants are able to suppress the preleukemic ones in a polyclonal mode, but not so in the monoclonal case, although the kinetic conditions are conserved.

The parameter values of the oncogenetic subpopulation can be tuned using the sliders. Thus, one can check for which parameter constellations one variant dominates over the other. The suppression of a subpopulation by the other one is contingent on the competition for resources. Generally, the growth of each clone,

*c*, obeys a logistic growth dynamics. Thereby, the carrying capacity,

_{i}(t)*k*, of each clone depends on the concentrations of the other clones as well as on their kinetic parameters. For example, the proliferation of T cells relies on the activation through self-peptides of the MHC (major histo-compatibility complex). Above all, the parameter values

_{i}*a*(avidity between clon

_{ij}*i*and resource

*j*) reflect the degree of competition between the T cells for these self-peptides. State of the art is the hypothesis, that each clone has a specific corresponding self-peptide, which in analogy to the ecosystem is called the clones "niche". However, additionally to the specific niches, all the T cell clones exhibit a (somewhat smaller) unspecific avidity with respect to the other peptides. The avidity matrix is created from a Gaussian distribution whose mean and standard deviation can be tuned via the sliders.

Normally, T cells are able to suppress oncogenetic variants, because in a healthy immune system they have a competitive edge on the oncogenetic cell variants. In monoclonal settings, however, the preleukemic variant may dominate. The presented model aims at suggestions of targeted experiments. In turn, the model should be adpated to future results of such experiments.