State-transition diagrams for biologists

Abstract

It is clearly in the tradition of biologists to conceptualize the dynamical evolution of biological systems in terms of state-transitions of biological objects. This paper is mainly concerned with (but obviously not limited too) the immunological branch of biology and shows how the adoption of UML (Unified Modeling Language) state-transition diagrams can ease the modeling, the understanding, the coding, the manipulation or the documentation of population-based immune software model generally defined as a set of ordinary differential equations (ODE), describing the evolution in time of populations of various biological objects. Moreover, that same UML adoption naturally entails a far from negligible representational economy since one graphical item of the diagram might have to be repeated in various places of the mathematical model. First, the main graphical elements of the UML state-transition diagram and how they can be mapped onto a corresponding ODE mathematical model are presented. Then, two already published immune models of thymocyte behavior and time evolution in the thymus, the first one originally conceived as an ODE population-based model whereas the second one as an agent-based one, are refactored and expressed in a state-transition form so as to make them much easier to understand and their respective code easier to access, to modify and run. As an illustrative proof, for any immunologist, it should be possible to understand faithfully enough what the two software models are supposed to reproduce and how they execute with no need to plunge into the Java or Fortran lines.

Figure 1. Three graphical illustrations of immunological knowledge which are very similar in spirit to state-transition diagrams: extract from Janeway’s classical immunology textbook illustrating the successive thymocyte differentiation stages - extract from a paper by Veronique Thomas-Vaslin et al.

illustrating the conveyor belt model of thymocyte differentiation to be discussed later - extract from a paper by Rong and Perelson illustrating the successive infection stages by the HIV virus.

Figure 2. Three classical elementary biological state transitions.

A cell is being infected by a virus, an inactive gene becomes active and a thymocyte switches between differentiation stages.

Figure 3. A more faithful and complete use of state-transition diagrams in the presence of origin and death states with possibility of proliferation and transit from stage A to B.

Figure 4. A state-transition diagram illustrating a conditional transition.

Here a DP cell can alternatively differentiate into either a SP4 or SP8 cell with respective probability and .

Figure 5. Example of compound state-transition diagram containing three single states and internal transitions.

A common transition leaving the compound state is shared by all single states although presenting different transition rates.

Figure 6. Two fully independent sub state-transition diagrams ( and ) executing in parallel.

Figure 7. Two sub state-transition diagrams ( and ) showing a weak dependency. The transition rates in the lower track depend on the state in the upper track.

Figure 8. Two sub state-transition diagrams ( and ) showing a strong dependency.

When the object is in state in the lower track, the transition becomes impossible. This is indicated by conditioning this transition with [NOT Y].

Figure 9. As a more realistic example of weak and strong dependencies, the figure illustrates three parallel sub-state-transition diagrams of a very classical immunological situation with lymphocytes differentiating, migrating and dividing (in 3 successive stages).

When entering its dividing cycle, all other transitions turn out to be impossible. This is indicated by the presence of [Not cycle] with labeling the compound state of dividing cell. Some of the transition rates “weakly” depend on the states where the lymphocyte is currently found in the other parallel tracks. indicates the birth rate to naive cells.

Figure 10. An illustration of a coupled transition (existing) in parallel tracks, representing a simple cell division and differentiation (left).

As cells transit from the cycling stage back to state , they should also move through the successive differentiated types - , and are three successive generations of cells. The default interpretation goes as the top-right figure but should rather be transformed into the bottom-right due to the shared labeling of the coupled transitions.

Figure 11. State-transition diagram of the Thomas-Vaslin et al. model of thymocyte differentiation.

This diagram attempts to essentially capture the same biological information as the one contained in figure 1b.

Figure 12. Statechart description of the Souza et al. model depicting differentiation, proliferation, TCR-MHC binding and movement between the different anatomical compartments of the thymus.

Although many transitions occur with parameterized rates, those related to differentiation are rule-based – making use of transition guards that refer to individual attributes. This model has no real hierarchical structure other than to conveniently specify a shared transition for SP4/SP8 cells.