^{1}, A. J. Pons

^{1}, N. Domedel-Puig

^{1}and J. García-Ojalvo

^{1,a)}

### Abstract

We investigate the dynamics of cell signaling using an experimentally based Boolean model of the human fibroblast signal transduction network. We determine via systematic numerical simulations the relaxation dynamics of the network in response to a constant set of inputs, both in the absence and in the presence of environmental fluctuations. We then study the network’s response to periodically modulated signals, uncovering different types of behaviors for different pairs of driven input and output nodes. The phenomena observed include low-pass, high-pass, and band-pass filtering of the input modulations, among other nontrivial responses, at frequencies around the relaxation frequency of the network. The results reveal that the dynamic response to the external modulation of biologically realistic signaling networks is versatile and robust to noise.

One of the characteristic features of living cells is their ability to continuously monitor their environment and respond appropriately to extracellular signals, which instruct the cells to take decisions such as proliferating, stopping growth, secreting chemicals, or even committing suicide. This behavior is mediated by signal transduction networks, which are composed of large numbers of interacting proteins. These networks have an input layer consisting of receptor proteins on the cell membrane that activate upon binding extracellular signals and an output layer of enzymes and/or transcription factors whose activation produces physiological changes in the cell. Until recently, the structure of these networks was largely unknown, and as a result, their theoretical study had to assume random connectivity between the nodes (proteins). Nowadays, the rise of high-throughput screening techniques allows the mapping of signaling networks with multiple connected pathways, and thus examining the activity of biologically realistic networks has become feasible. Here, we study the dynamical behavior of a signaling network recently identified in human fibroblasts in terms of a Boolean description in which the network elements are either fully active or inactive. Using extensive and systematic numerical simulations, we quantify the response of the network to all its inputs, and especially the relaxation dynamics to the corresponding stationary attractors, both with and without noise in the input signals. We also examine the case of periodically modulated inputs and characterize the frequency response of the network, which is shown to be extremely diverse.

This work has been financially supported by the Spanish Network of Multiple Sclerosis (REEM, Instituto de Salud Carlos III, Spain), the Fundación Mutua Madrileña (Spain), the Ministerio de Ciencia e Innovación (Spain, Project No. FIS2009-13360 and I3 program), and the Generalitat de Catalunya (Project No. 2009SGR1168). P.R. is supported by an FI grant from the Generalitat de Catalunya. A.J.P. was supported by the Juan de la Cierva program of the Ministerio de Ciencia e Innovación (Spain). J.G.-O. also acknowledges financial support from the ICREA Foundation.

I. INTRODUCTION

II. THE BOOLEAN NETWORK MODEL

III. RESULTS

A. Constant inputs

B. Inputs with constant chatter level

C. Inputs with constant chatter level plus a periodic input signal

IV. DISCUSSION

### Key Topics

- Networks
- 19.0
- Attractors
- 18.0
- Signal transduction networks
- 16.0
- Intracellular signaling
- 15.0
- Proteins
- 7.0

## Figures

The network model used in this work consists of 130 internal nodes and 9 input nodes, which represent protein species being either active (1) or inactive (0), connected through 542 interactions. Input signals (top) and output nodes (bottom) are labeled with abbreviations that are defined in the text.

The network model used in this work consists of 130 internal nodes and 9 input nodes, which represent protein species being either active (1) or inactive (0), connected through 542 interactions. Input signals (top) and output nodes (bottom) are labeled with abbreviations that are defined in the text.

Dynamics of the system for constant inputs. (a) Dynamic evolution of the network nodes for random initial conditions and a fixed combination of input states (stress, IL1/TNF, ECM, and are inactive, while pump, EGF, , , and are active). White (black) regions correspond to active (inactive) states. After some initial transient, the network reaches a limit cycle attractor where the activity of some of nodes oscillate periodically, while that of the others is fixed to either the active or inactive state. (b) Histogram of the transient lengths for different random initial conditions (see main text). Transient lengths were computed by first finding the attractors and then identifying the iteration at which the attractor is reached. The inset in (b) shows the histogram in doubly logarithmic axes, which reveals that the tail of the histogram follows a power law (red dashed line) with scaling exponent 2.4. Note that the data shown in the inset has not been divided by .

Dynamics of the system for constant inputs. (a) Dynamic evolution of the network nodes for random initial conditions and a fixed combination of input states (stress, IL1/TNF, ECM, and are inactive, while pump, EGF, , , and are active). White (black) regions correspond to active (inactive) states. After some initial transient, the network reaches a limit cycle attractor where the activity of some of nodes oscillate periodically, while that of the others is fixed to either the active or inactive state. (b) Histogram of the transient lengths for different random initial conditions (see main text). Transient lengths were computed by first finding the attractors and then identifying the iteration at which the attractor is reached. The inset in (b) shows the histogram in doubly logarithmic axes, which reveals that the tail of the histogram follows a power law (red dashed line) with scaling exponent 2.4. Note that the data shown in the inset has not been divided by .

Population dynamics of the system for constant inputs. (a) Population average for output nodes obtained from 3000 realizations of the network dynamics with different initial conditions and the same fixed combination of input states as in Fig. 2(a). Different combinations of fixed inputs lead to different population averages of the outputs. (b) Histogram of the transient lengths of the population dynamics for the 512 possible combinations of input states. Transients for the population dynamics are computed as the maximum transient lengths of the average activities of single nodes. These node-specific transients were calculated by first establishing the range of values of the stationary state (within a certain tolerance) and then finding the iteration at which the node’s average activity value enters this range and never leaves it again. The population relaxation durations are close to the median transient lengths of individual realizations of the network dynamics [cf. Fig. 2(b)]. As in Fig. 2(b), the inset in plot (b) depicts the histogram in doubly logarithmic scale. The plot indicates that this histogram has again a fat tail, although in this case the number of data points is too small to reveal a clear-cut power law.

Population dynamics of the system for constant inputs. (a) Population average for output nodes obtained from 3000 realizations of the network dynamics with different initial conditions and the same fixed combination of input states as in Fig. 2(a). Different combinations of fixed inputs lead to different population averages of the outputs. (b) Histogram of the transient lengths of the population dynamics for the 512 possible combinations of input states. Transients for the population dynamics are computed as the maximum transient lengths of the average activities of single nodes. These node-specific transients were calculated by first establishing the range of values of the stationary state (within a certain tolerance) and then finding the iteration at which the node’s average activity value enters this range and never leaves it again. The population relaxation durations are close to the median transient lengths of individual realizations of the network dynamics [cf. Fig. 2(b)]. As in Fig. 2(b), the inset in plot (b) depicts the histogram in doubly logarithmic scale. The plot indicates that this histogram has again a fat tail, although in this case the number of data points is too small to reveal a clear-cut power law.

Effects of stochastic chatter acting upon the input nodes. (a) Dynamic evolution of the network nodes for a given realization of chatter affecting all input nodes. In this case, the evolution of the network is erratic and does not reach stable or periodic attractors [compare the evolution shown here with that presented in Fig. 2(a)]. (b) Population average for the output nodes [color coding as in Fig. 3(a)] obtained from the 3000 realizations of the network dynamics for a fixed value of chatter for all input states.

Effects of stochastic chatter acting upon the input nodes. (a) Dynamic evolution of the network nodes for a given realization of chatter affecting all input nodes. In this case, the evolution of the network is erratic and does not reach stable or periodic attractors [compare the evolution shown here with that presented in Fig. 2(a)]. (b) Population average for the output nodes [color coding as in Fig. 3(a)] obtained from the 3000 realizations of the network dynamics for a fixed value of chatter for all input states.

Response of the network to periodic driving. Power spectrum of the average activity of an output node (Akt) vs stimulation frequency, , being applied to a given input node (stress), while the other input nodes are subject to a fixed chatter level .

Response of the network to periodic driving. Power spectrum of the average activity of an output node (Akt) vs stimulation frequency, , being applied to a given input node (stress), while the other input nodes are subject to a fixed chatter level .

Frequency filtering of the network. Power spectral density of the average output-node activity at the input frequency , normalized to the power of the input signal, as a function of , for one input node and fixed chatter level for the rest of inputs. The periodic driving is applied to (a) stress and (b) EGF input nodes.

Frequency filtering of the network. Power spectral density of the average output-node activity at the input frequency , normalized to the power of the input signal, as a function of , for one input node and fixed chatter level for the rest of inputs. The periodic driving is applied to (a) stress and (b) EGF input nodes.

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