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### attractors and cellular states

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Posted by ilya_shmulevich
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**22 Jan 2010 **at** 06:55 GMT**

In this paper, the authors argue against what they call the "attractor hypothesis," which asserts that the stable states of the biomolecular system, modeled as a high-dimensional dynamical system, correspond to functional cellular states or fates. The process of a cell changing states (e.g., upon differentiation) corresponds to a state trajectory through the state space of the dynamical system toward such a stable state.

I find the authors' rationale for discounting what they call the "state space approach", by which I presume they mean simply the dynamical systems approach for modeling biomolecular networks, rather confusing. In the Introduction, they state that one of the advantages of this approach is that it provides a framework for predictive modeling. And yet in the Author Summary, they state that the problem with this approach is that it lacks predictive power. How does one resolve this contradiction? Furthermore, the authors seem to use the terminology "state space", which they endow with quotation marks, merely as a linguistic metaphor, as if this is a concept that is devoid of a precise mathematical definition.

This raises the question of whether the authors are denying the fact that biomolecular systems, regardless of how they are modeled mathematically, contain multiple (meta-)stable steady states, referred to as attractors of the underlying dynamical system. If not, then how does one interpret such steady states at the functional (cellular) level? Put simply, what is the cell doing when it finds itself in one of its steady states (as opposed to undergoing transient state changes, such as after receiving a stimulus)?

It is absolutely erroneous to state that "Each observed phenotype can be represented as a single point in the state space" -- such a view not only ignores stochastic effects due to intrinsic and extrinsic noise, and consequentially, population heterogeneity if the phenotype is determined on a multi-cellular level, but is inadequate as most functional cellular states, such as the cell cycle, exhibit stable and highly robust dynamics that clearly cannot be captured by a single point in the state space of the dynamical system. Indeed, the dynamics, stability, and robustness of the cell cycle attractor have been studied in several organisms, e.g., [1, 2, 3].

The main argument in favor of evading the existence of attractors appears to be that it limits mechanistic interpretations or construction of predictive models (see last paragraph of Introduction). And yet, nowhere is this justified or logically argued. The conclusion that cell fates are not attractors simply appears to be a non sequitur. Furthermore, it is not at all clear how the method described in this paper, with its two groups of genes, is predictive. What does it predict and how are the two groups of genes different from an arbitrary classification?

Rather, the authors offer "an alternative, more plausible interpretation," though it is not clear of what, that rests on notions such as "core biological processes", "transient processes", and "activated pathways" (this time the quotations are my own), which, upon careful examination, are not inter-subjective definitions. Taking but one representative sentence from the paper -- "...the core and transient components suggested by our trajectory model can be thought of as being a “core” component equivalent to the central pathway between states with the transient components representing orthogonal perturbations relative to the core downhill pathways" -- illustrates the difficulty of interpreting such terms unambiguously.

In contrast, dynamical systems theory, which formalizes the notions of state trajectories and attractors, and sets the foundation for engineering approaches, such as control systems (in the context of optimal therapeutic intervention strategies), was recognized to be of central importance in biology already by Norbert Wiener, who became "aware of the essential unity of the set of problems centering about communication, control, and statistical mechanics, whether in the machine or in living tissue" [4].

In summary, it is entirely unclear how anything reported in this paper contradicts the idea of an attractor.

Ilya Shmulevich

Institute for Systems Biology

[1] Proc Natl Acad Sci U S A. 2004 Apr 6;101(14):4781-6.

[2] PLoS One. 2008 Feb 27;3(2):e1672.

[3] Bioinformatics. 2006 Jul 15;22(14):e124-31.

[4] Wiener, N. (1948). Cybernetics; or, Control and communication in the animal and the machine. [Cambridge, Mass.]: Technology Press.

**No competing interests declared.**

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attractors and cellular states

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