^{1}

^{2}

^{3}

^{4}

^{1}

^{1}

^{3}

^{*}

^{1}

^{2}

^{3}

^{4}

Edited by: Tomáš Helikar, University of Nebraska-Lincoln, United States

Reviewed by: Jongrae Kim, University of Leeds, United Kingdom; Anatoly Sorokin, Institute of Cell Biophysics (RAS), Russia

This article was submitted to Systems Biology, a section of the journal Frontiers in Physiology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Logical models offer a simple but powerful means to understand the complex dynamics of biochemical regulation, without the need to estimate kinetic parameters. However, even simple automata components can lead to collective dynamics that are computationally intractable when aggregated into networks. In previous work we demonstrated that automata network models of biochemical regulation are highly canalizing, whereby many variable states and their groupings are redundant (Marques-Pita and Rocha,

Mathematical and computational modeling of biological networks promises to uncover the fundamental principles of living systems in an integrative manner (Iyengar,

Two reasons contribute to the success of BN models: (i) the reduction of complex multivariate dynamics to a graph revealing the organization and constraints of the topology of interactions in biological systems, and (ii) a coarse-grained treatment of dynamics that facilitates predictions of limiting behavior and robustness (Bornholdt,

Here we present CANA^{1}

Here we demonstrate the full functionality of the CANA package using the BN model of floral organ development in the flowering plant

The CANA package fills a key void in the available library of computational software to analyze Boolean Network models. Existing software falls into two categories: either they are designed to reverse engineer BN models from biological experimental data, or they focus on simulating BN dynamics. Examples of the first category include the

A ^{k} → {0, 1}, is defined by a ^{k} combinations of input states and a mapping to the automaton's next state (transition or output), ^{t+1} (Figure

CANA analysis of the Boolean automaton defining the dynamics of the TFL1 gene in the BN model of the floral organ arrangement in the flowering plant _{4}≡{_{5}, _{6}}, where the input variable AP2 can be either _{r}), input symmetry (_{s}), and effective connectivity (_{e}) of TFL1 automaton. Values in parenthesis are the respective (relative) measures normalized by

A _{i} ∈ _{ji} ∈ _{i}, _{j} ∈ _{j} is an input to automaton _{i}, as computed by _{i}. The set of inputs for automaton _{i} is denoted by _{i} = {_{j} ∈ _{ji} ∈ _{i} = |_{i}|, is the _{i}. At any given time ^{t}) for the collection of states of all automata of the BN at time ^{0}, by a _{α} at time _{β} at time _{α, β} is allowed out of every configuration node _{α}. Configurations that repeat, such that

In CANA, a python class named ^{2}

Important insights about BN dynamics are gained by observing that not all inputs to an automaton are equally important for determining its state transitions, a concept known as

Several measures of canalization present in the LUT of an automaton are also defined in CANA, and can be accessed by function calls to both the _{r}(_{e}, is a complementary measure of _{r}(_{e}(_{s}(

where ^{3}

Most automata contain redundancy of one or both of the two forms of canalization; only the two parity functions for any _{r} = 0 (e.g., the _{s} > 0. Therefore, the original interaction graph of a BN tends to have much redundancy and does not capture how automata truly influence one another in a BN. To formalize this idea, the CANA package computes an _{ji} ∈ [0, 1]∀_{i}, _{j} ∈ _{j} in determining the truth value of automaton _{i}, and computed via Equation 2. Specifically, we define per-input measures of canalization for

where (_{υ} is a logical condition that assumes the truth value 1(0) if input _{θ} for a position-free symbol in schema

The effective graph was shown to be important in predicting the controllability of BN (Gates and Rocha, _{e} of BN (the mean in-degree of the effective graph) is a better predictor of criticality than the in-degree of the original interaction graph (Manicka, _{j} to automaton _{i} exists in the original interaction graph _{ji} = 1, but not in the effective graph _{ji} = 0, because it is fully redundant and does not affect the automaton's transition (see several such cases in Figures

BN model of the floral organ arrangement in the flowering plant _{ji} (Equation 2). Some edges, originally in

The canalizing logic of an automaton provided by the schemata set _{i} = 0, white, or _{i} = 1, black), and

The discovery of control strategies in BN models is a central problem in systems biology; theoretical insights about controllability can enhance experimental turnover by focusing experimental interventions on genes and proteins more likely to result in the desired phenotype output. It is well-known that when the set of automata nodes

CANA contains Python functions designed to provide a testbed for the development of BN control strategies, and to investigate the interplay between canalization, control, and other dynamics properties. Specifically, we study the control exerted on the dynamics of a BN, ^{|D|}−1 configurations in the STG, which are reachable given the bit-flip perturbations of the driver variables. A BN is controllable when every configuration is reachable from every other configuration in

CANA computes the CSTG of

where, for each configuration _{α}, _{β} lying on all directed paths from _{α}, normalized by the total number of other configurations 2^{N−1}. Similarly, the

In Systems Biology applications, typically only the attractors of BN are meaningful configurations, used to represent different cell types (Kauffman, _{κ} to attractor _{γ} in the CSTG

where

_{1}, …, A_{10} represents an attractor of the network dynamics. The BN configurations for steady-state attractors A_{3} and A_{5} are shown as interaction graphs with node variables colored white or black for states _{i} = 0 and _{i} = 1, respectively; driver variables are shown with a yellow contour.

Finally, CANA also provides the functionality to approximate the minimal driver variable subset using two prominent network control methodologies:

We presented a novel, open-source and publicly-available software platform that integrates the analytic methodology used to study canalization in automata network dynamics. This methodology can now be used by others to simplify large automata networks, especially those in models of biochemical regulation dynamics. In addition to the extraction and visualization of specific effective pathways that regulate key phenotypic outcomes in a sea of redundant interaction, CANA includes functionality to measure canalization, uncover control variables, and study dynamical modularity, robustness, and criticality. We hope that the consolidation of redundancy and control algorithms into one package encourages other researchers to build upon our work on canalization, thus adding additional algorithms to CANA.

The CANA python package and all datasets analyzed for this study can be found on Github at

RC, AG, and XW contributed to the CANA package. LR developed the per-input measures of canalization and the effective graph formulation. RC, AG, and LR wrote the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We would like to thank Manuel Marques-Pita, Santosh Manicka, and Etienne Nzabarushimana for helpful conversations throughout the development of the CANA package.

^{1}

^{2}Future releases will provide a direct link to the Cell Collective API for conversion of Cell Collective models. Currently, models are converted to .CNET (truth table) format, and subsequently imported to CANA.

^{3}_{r} and _{e} can be computed on either set of schemata _{s} must be computed on