Project

Quantitative methods in systems immunology

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Scientific background

Experimental data in biology and medicine are getting increasingly quantitative calling for the development of quantitative models of biological systems. Suitable methodologies can be borrowed from a wide range of methods employed in physics for the description natural science problems. This is what we do.

Rather than using our expertise in a specific methodology and applying this method to a diversity of problems in life science, we decided always to start from an interesting question in the life sciences and to search for the appropriate method to answer it.

As we are mostly interested in systems with particular dynamics and spatial organization, this naturally leads to spatio-temporal modelling approaches. In some cases, this requires the development of new methods.

Discrete agent-based models

When rare events are important for the problem under consideration, stochastic event generators are appropriate. Such state-space models can be complemented by explicit discretized representations of space in two or three dimensions. We use this grid for cell or organelle motility as well as for solving reaction-diffusion systems. Each cell or organelle is represented as an instance of a class in C++. Each object evolves its internal state, secretes, senses and consumes solubles, move in space (eventually in dependence on chemokine gradients).

It is this kind of hybrid event generators that were employed for the germinal centre research (including the LEDA model) and for some analyses of multi-photon lymphocyte migration data. We extended this method to a sub-cellular representation with an amorphous cytosol. This rather simple sub-cellular model was able to grasp many parameters of lymphocyte motility. We called this modelling platform of interacting cells with a subcellular resolution HYPHASMA, greek for tissue.

  • Meyer-Hermann M, Maini PK. Interpreting two-photon imaging data of lymphocyte motility. Phys Rev E 71 (2005) 061912-1-12.

Lattice-free agent-based models and Delaunay-Object-Dynamics

Intracellular and cellular models may require the representation of individual agents in a lattice-free environment. We are using molecular dynamics derived methods with interactions between agents defined by interaction potentials. Lenard-Jones potentials are suitable for cell-cell interactions because they inherently include hard-core repulsion and adhesion.

As for the lattice-based models, these models may be combined with a reaction-diffusion system which is solved on an overlaid grid. We applied this kind of approach to problems in biofilm formation, cancer, and dynamics of intracellular organelles.

The observation that densely packed cells acquire a polygonal shape initiated the development of a new modelling strategy that combines cell geometry with the philosophy of molecular dynamics. This modelling platform is called Delaunay-Object-Dynamics (DOD). In DOD every cell is represented as a node. Its neighbours are derived from the Delaunay triangulation of all nodes using the empty circumsphere criterion. This replaces the sometimes difficult search for interaction neighbours in molecular dynamics because in DOD, the interaction neighbours are unambiguously determined. The dual of the Delaunay triangulation is the Voronoi tessellation, which gives every cell object a shape, a size, and an interaction face to other cells. The size of this interaction face is also known by construction and enables application of classical Newton mechanics.

If the biomechanical parameters of the cells are known, this model is a quantitative representation of cell dynamics with a high level of predictive power. We have made this concept usable for biological problems with many cells by developing a sequential and parallel version of a code with local adaptation of the Delaunay triangulation rather than re-triangulation of the whole system.

Delaunay triangulation of vertices N1-k and Voronoi tessellation (red area) as defined by the empty circumsphere criterion in 2D (left). M1-k are the centres of the circumspheres. In 3D (right), a…

  • Grise G, Meyer-Hermann M. Surface reconstruction using Delaunay triangulation for applications in life sciences, Comput Phys Commun 182 (2011) 967-977.
  • Grise G, Meyer-Hermann M. Towards sub-cellular modeling with Delaunay triangulation, Math Model Nat Phenom 5 (2010) 224-238.
  • Meyer-Hermann M. Delaunay-Object-Dynamics: Cell mechanics with a 3D kinetic and dynamic weighted Delaunay-triangulation. Curr Top Dev Biol 81 (2008) 373-399.
  • Schaller G, Meyer-Hermann M. Multicellular tumor spheroid in an off-lattice Voronoi/Delaunay cell model. Phys Rev E 71 (2005) 051910-1-16.
  • Beyer T, Schaller G, Deutsch A, Meyer-Hermann M. Parallel dynamic and kinetic regular triangulation in three dimensions. Comput Phys Commun 172 (2005) 86-108.
  • Schaller G, Meyer-Hermann M. Kinetic and dynamic Delaunay tetrahedralizations in three dimensions. Comput Phys Commun 162 (2004) 9-23.

Differential equations, parameter fitting, and identifiability analysis

When a system is composed of many identical and well intermixed entities, differential equations are a good choice. For example, the dynamics of specific factors in blood samples are well described by ordinary differential equations. The big advantage of this methodology is that they are easily solved and solutions may even be found analytically with pencil and paper. We also prefer differential equations for modelling more complex molecular networks over alternative approaches, like boolean networks, which involve less parameters but often lead to artificial solutions because natura not facit saltum (Gottfried Leibniz).

Given a model and a set of experimental data points, the model parameters are not always known. We take care that all our models are formulated in a language in which every parameter has a physiological meaning and is, thus, in principle measurable. In some cases, the measurements are difficult. Then the remaining unknown parameters have to be determined by reverse engineering: The read-out of the model has to fit the data. It is important to realize that not every data set contains information on the unknown model parameters. This can be determined by an identifiability analysis. Correlations between parameters are determined by bootstrapping based on refitting with artificially modified data sets.

In order to fit model parameters, a measure for the quality of the model read-out is defined based on deviations of the model output from the data. This measure is minimized by scanning the parameter space. In most cases, the absolute minimum of the measure is not found by standard or stochastic difference methods. Therefore, we prefer to use differential evolution algorithms. When the model gets too complex, finding an acceptable best fit might still be impossible.

We have invented an iterative method with which the complex problem is decomposed into simpler subsystems. Each of these subsystems is solvable by differential evolution algorithms. The, thus, generated approximate fit of the total system is then used as a starting point for a global fit. This method outperformed the unmodified global fit in all network topologies tested so far.

  • Binder SC, Hernandez-Vargas EA, Meyer-Hermann M. Reducing complexity: an iterative strategy applied to parameter selection for regulatory networks. Computer Physics Communications 190 (2015) 15-22.

Investigators:

  • Sebastian C. Binder
  • Jaber Dehghany
  • Michael Meyer-Hermann

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