What is a Cell Cycle Checkpoint? The TotemBioNet Answer

Abstract

TotemBioNet is a new software platform to assist the design of qualitative regulatory network models by combining “genetically modified Hoare logic”, temporal logic model checking and optimized enumeration techniques. TotemBioNet is particularly efficient to manage parameter identification, the most critical step of formal modelling. It is also remarkably flexible and efficient to check properties in order to explore biological assumptions. To illustrate this efficacy, we address the classical example of the cell cycle, where the passage from one phase to the next one, called checkpoint, is crucial but is usually a rather fuzzy informal concept. The cyclic behaviour of the cell cycle is specified by temporal logic and the order of individual events inside each phase is explored thanks to quantifiers introduced in Hoare logic. This way, TotemBioNet rapidly suggests a sensible formalization of the notion of checkpoint.

Appendices

Appendix A: Static Description of the Cell Cycle Model

See Fig. 2.

Fig. 2.

A 5-variable interaction graph of the mammalian cell cycle, from [2]. Progression through the cell cycle is driven by 2 types of genetic entities: complexes of Cyclins/Cyclin-dependant kinases (Cyc/Cdks) and their inhibitors known as ennemies. The 5 variables of the graph represent these entities, in orange. sk is the abstraction of both complexes CycE/Cdk2 and CycH/Cdk7, known as starting kinases. a and b respectively represent CycA/Cdk1 and CycB/Cdk1. en is the abstraction of the main Cyc/Cdks ennemies: the anaphase-promoting complex APC/Cdh1, cyclin-kinase inhibitors p21 and p27, and Wee1 protein. The variable ep is the anaphase-promoting complex APC/Cdc20, which is a Cyc/Cdks ennemy involved in mitosis exit and so-called exit protein. Regulations between variables are described in [2]. This interaction graph was designed using the tool yEd (www.yworks.com/products/yed). (Color figure online)

Appendix B: Equivalent Specification of $H_{init}$ using a Fair CTL Formula

In the first experiment, the cell cycle is specified by the $H_{init}$ Hoare triple. Here, the cell cycle is specified by the $\varphi _{init}$ CTL formula depicting the $H_{init}$ path.

Appendix C: Specification of $H_{perm}$

This Hoare triple encodes a cell cycle in which phases are described by all permutations of their respective transitions. G1 is specified in blue, S in red, G2 in grey and M in green.

Appendix D: Specification of $H_{permG1}$

This Hoare triple describes the cell cycle in which G1 in addition to G2 allows all permutations of its transitions.

Appendix E: Specification of $\varphi _{G1/S}$ with CTL

The premise $G1_{init}$ of the formula $\varphi _{G1/S}$ is the precondition of the Hoare triple $H_{init}$ defined in [2]. It defines the initial state of G1. The first transition of S, $a+$, must not occur before any G1 transition. Thus 9 paths must not exist starting from the first G1 state encoded in premise.

The notation $EX(a=1 \wedge EX(sk=1 \wedge EX(en=0 \wedge EX(sk=2))$ is a CTL version of the Hoare path: $a+; sk+; en-; sk+$.

Appendix F: Specification and Checking of S/G2 and M/G1 Checkpoints with CTL

The premise of $\varphi _{S/G2}$ formula (see below) encodes the first state of S. $a-$ and $ep+$ are the 2 possible first events of G2 according to $H_{permG1}$. They must not occur before completion of S events. Thus 21 paths must not exist starting from the state in premise. $\varphi _{S/G2}$ is then defined as:

Similarly, the premise of $\varphi _{M/G1}$ formula (see below) encodes the first state of M. $sk+$ and $en-$ are the 2 possible first events of G1 according to $H_{permG1}$. Thus the 8 paths enabling these events to occur before completion of M events must not exist, starting from the state in premise. $\varphi _{M/G1}$ is then defined as:

TotemBioNet extracts no model (Table 2) for each of these checkpoints, from which we conclude that the model is not precise enough to capture them.

Table 2. Verification of S/G2 and M/G1 checkpoints. $H_m$ is the set of models satisfying Hoare and Snoussi constraints. $S_m$ is the set of selected models after model-checking of a temporal logic formula on each element of $H_m$.