## Final Report Summary - RUBYX (Intrinsic Complexity of Rule-Based Systems)

The main results of RUBYX fall into two categories.

Fundamental mathematical results

RUBYX fortifies the mathematical foundations of rule-based modeling of bio-chemical reaction networks, social networks, and, more generally, any stochastic process with graph-like states that evolves by random, local changes of connectivity.

Feasible methods and algorithms

RUBYX describes a non-trivial class of systems whose time evolution can be predicted efficiently, which includes the preferential attachment process from the pioneering days of social network theory.

Fundamental Mathematical Results

The main result of RUBYX is a mathematical characterization of the time evolution of the number of molecules or molecule fragments of a specific type in idealized bio-chemical reaction networks--formalized by expected motif counts in rule-based models conditioned on possible initial states: it can be obtained as the unique solution of an abstract evolution equation in a suitably chosen infinite-dimensional vector space, under conditions that ensure at most exponential growth over time. Moreover, the time evolution of the expected number of molecules can be computed for typical models of practical relevance, up to arbitrary desired precision, using only partial information about the initial distribution. Thus, RUBYX has established the mathematical and algorithmic foundations for a novel analysis technique of rule-based models of stochastic systems, as used in synthetic biology, theoretical physics, and complex systems.

The main novelty is that, in contrast to the vast body of work on stochastic simulation, the system state is either fixed or abstracted away from completely, which becomes possible by instead taking observables to evolve in time. We are thus arriving at a fundamental analogy to the Heisenberg picture for quantum mechanics--again, keeping the system state fixed while the observation is varying in time--based on a theorem for stochastic processes, which calls for a subtle change in the underlying mathematics in comparison to Quantum mechanics, due to the absence of an inner product.

The fundamental characterization of trajectories of observables in time as unique solution to abstract evolution equations opens access to a wide range of numerical methods to compute the latter, thus putting to use recent advances in Operations research via fundamental results in theoretical computer science, concerning the limits of the computable. However, the main insight at the very end of the project RUBYX are unexpected phenomena in the envisaged quantitative analysis of rule-based models, indicating that predictability of systems might be beyond the computability barrier for certain models that involve the duplication of entities and their neighborhood structure. The rough idea is that the change of expected motif counts might be varying too fast to be traceable, even theoretically. This shows the flip side of the immense expressive power of rule-based modeling, namely difficulties or even impossibility to analyze or “solve” these models.

Feasible Methods and Algorithms

Besides the principal theoretical result, namely computability of the time evolution of expected motif counts in rule-based systems, RUBYX has identified a well-behaved class of rule-based systems--exemplified by the preferential attachment process, which is an extremely popular model from the early days of network modeling--where the prediction of the time evolution of mean motif counts is feasible. This is done by transforming a given rule-based model with a certain motif of interest into an ordinary linear differential equation with a finite number of variables, which subsequently can be solved by one of the various numerical methods from the literature, which are implemented in freely available software such as GNU Octave. The computed solution then describes the time evolution of the expected number of motif occurrences.

One interesting theoretical aspect, based on a standard example in text books on stochastic processes, the so-called Yule-Furry process, is that a tight classification of the intrinsic complexity of rule-based systems leads to a longstanding open problem in the complexity theory for real valued functions, namely tight lower bounds for how difficult it is to approximate the exponential function f(x)= e^x. Thus, even the simplest rule-based models lead to extremely subtle theoretical questions.

Conclusion

The project RUBYX has found unanticipated facts concerning the computability of the time evolution of rule-based systems, showing that, in general, they are much more intricate than the success stories for the analysis of specific models had indicated. Thanks to recent progress in the area of abstract evolution equations, RUBYX has made an important connection to the corresponding field of numerical mathematics, thus providing the latter research field with a wide range of important examples. At the same time RUBYX establishes the mathematical foundation of rule-based models used in chemistry, biology, and complex systems. Finally, also the question for the intrinsic complexity of rule-based models is partially answered for systems that can be transformed to ordinary differential equations. However, in the end, we are led to a longstanding open problem about the exponential function, the solution of a 1-dimensional linear differential equation, for the simplest type of systems and a conjecture that rule-based systems are not in general amenable to automatic analysis.

Socio-economic impact

Theoretical results typically do not have any direct impact on society and economics. However, RUBYX reinforces the trend of mechanistic conceptualization of human artifacts and natural phenomena. For example, by strengthening the mathematical foundations of rule-based modeling--a new paradigm for modeling stochastic processes that has been very successful in the area of synthetic biology, in particular in coping with the complexity of signaling pathways--RUBYX contributes indirectly to the field of executable models of larger and larger parts of the bio-chemical reaction networks in human cells, which in turn has promising applications in the pharmaceutical industry and the medical sector.

More generally, rule-based modeling can be understood as a programming paradigm for stochastic processes in continuous time, which are ubiquitous in modeling cyberphysical systems, telecommunication networks, and epidemic processes. Though there is no direct, objectively measurable impact, by providing a novel perspective on rule-based models--which is even in analogy to standard techniques for conventional programs if we consider observables as generalized “fuzzy” properties--RUBYX encourages a wider community of scientists and science enthusiasts to engage in the endeavor to model and understand inherently stochastic phenomena.