^{1}

^{2}

^{2}

^{3}

^{2}

^{*}

Conceived and designed the experiments: XZ MX. Performed the experiments: XG. Analyzed the data: MX HX. Contributed reagents/materials/analysis tools: XZ. Wrote the paper: XZ MX FA.

The authors have declared that no competing interests exist.

Diseases are believed to arise from dysregulation of biological systems (pathways) perturbed by environmental triggers. Biological systems as a whole are not just the sum of their components, rather ever-changing, complex and dynamic systems over time in response to internal and external perturbation. In the past, biologists have mainly focused on studying either functions of isolated genes or steady-states of small biological pathways. However, it is systems dynamics that play an essential role in giving rise to cellular function/dysfunction which cause diseases, such as growth, differentiation, division and apoptosis. Biological phenomena of the entire organism are not only determined by steady-state characteristics of the biological systems, but also by intrinsic dynamic properties of biological systems, including stability, transient-response, and controllability, which determine how the systems maintain their functions and performance under a broad range of random internal and external perturbations. As a proof of principle, we examine signal transduction pathways and genetic regulatory pathways as biological systems. We employ widely used state-space equations in systems science to model biological systems, and use expectation-maximization (EM) algorithms and Kalman filter to estimate the parameters in the models. We apply the developed state-space models to human fibroblasts obtained from the autoimmune fibrosing disease, scleroderma, and then perform dynamic analysis of partial TGF-β pathway in both normal and scleroderma fibroblasts stimulated by silica. We find that TGF-β pathway under perturbation of silica shows significant differences in dynamic properties between normal and scleroderma fibroblasts. Our findings may open a new avenue in exploring the functions of cells and mechanism operative in disease development.

Identifying differentially expressed genes across distinct conditions and clustering co-expressed genes into different functional groups have been general approaches for unraveling molecular mechanisms involved in disease pathogenesis

Another approach to deciphering pathogenesis of complex diseases is system thinking. Human complex diseases are believed to arise from malfunction of a specific biological system, rather than from isolated events. It is increasingly recognized that biological systems as a whole are not just the sum of their components but, rather, ever-changing, complex, interacted and dynamic systems over time in response to internal events and environmental stimuli

The dynamic behavior of most complex biological systems emerges from the orchestrated activity of many components (e.g. genes and proteins) that interact with each other to form complicated biological networks involving gene regulation and signal transduction

Investigation of dynamic properties of gene networks has three major tasks: development of mathematic models, estimation of the parameters in the models, and dynamic analysis. Mathematical modeling is to use mathematical language to describe the dynamic characteristics of a system

Another important property of the dynamic systems is the transient response to disturbance of internal noises and external environmental forces, which measures how fast the systems respond to the perturbation and characterizes damping and oscillation properties of the process in response to the perturbation

In summary, stability, transient response, feedback and controllability are basic dynamic properties of the biological systems and are essential to the function of the cells and organisms. As a proof of principle, in this report we investigate the differential dynamic properties of TGF-β pathway in response to perturbation of silica between normal and scleroderma fibroblasts. Scleroderma or systemic sclerosis (SSc) is a typical complex disease in which fibrosis occurs in multiple organs. Although etiopathogenesis is unknown, both genetic and environmental factors are believed to play critical roles. The major source of fibrosis in SSc is over production of collagens from fibroblasts. Fibroblasts obtained from SSc patients appeared to be genetically engineered to produce more collagens and cytokines

Gene regulation involves a large number of biochemical events. Although kinetic models can be developed for gene regulation

Previously, we found that the SPARC (secreted protein, acidic, and rich in cysteine) gene is involved in the regulation of extracellular matrix genes such as _{k}_{k}_{k}

We performed experiments on cultured human fibroblasts. We have 5 pairs each of normal and SSc patients' samples. For each sample, we have two replications were perturbed by Silicon. The transcript levels of six genes: _{1}_{2}_{3}_{4}_{5}_{1}_{2}

A graph will be used to represent a genetic network. The nodes in the graph will represent the variables that correspond to the expressions of the genes. The edge between two nodes denotes that two variables are dependent. The number next to edges is the elements of the parameter matrices A, B, C and D in the state-space model. The estimated state-space model is shown in

The numbers next to the edges are the coefficients in the state-space equations for the normal (black color) and SSc (red color) fibroblasts, respectively. The numbers in the boxes denote the mean expression values of the genes in normal (black color) and SSc (red color) fibroblasts.

The most important dynamic property of gene regulatory networks is concerned with stability. Stability is an organizing principle of gene regulatory networks

The eigenvalues of the transition matrix

Normal fibroblasts | SSc fibroblasts |

0.23997 | 1.10627 |

0.66237 | 1.46736 |

−0.24207 | 1.00586 |

0.84603 | 0.63654 |

0.32847 | 0.78374 |

The dynamic behavior of a system is encoded in the temporal evolution of its states. Cell functions are essentially temporary processes and largely determined by the dynamic properties of the biological systems in the cells. Transient and steady state responses are two steps of the response of a gene network to perturbation of external environments. The transient response of the gene network to perturbation of environments is defined as rapid changes of the expressions of the genes in the network over time which go from their initial states to final states after perturbation of external input

Popular methods for investigation of the transient responses of the dynamic systems are to study the time domain characteristics of the system under perturbation of the external signals. The transient response of the dynamic system depends on the input signal. Different input signal will lead to the different transient response of the system. There are numerous types of signal in practice. For the convenience of analysis and comparison, we consider two types of signals: (i) unit-step signal and (ii) unit-impulse signal as shown in the

For discrete dynamic systems, the transient response of the system is obtained by using the inverse

The performance of the gene networks under the design for stability, time response and reliability can be studied by analysis of their corresponding closed-loop system. The basic features of the stability and transient response of the closed-loop system are largely determined by the location of the closed-loop poles, which in turn is related to the value of the loop gain _{1} may become unstable for a different gain K_{2}. We often observe that the root-locus will move from the left-hand of the s-plane to the right-hand of the s-plane, which implies that stable system becomes unstable as the system gain changes from one region to another region.

The root-locus plots the locations of the poles of the closed-loop single input and single output system (SISO) as the system gain varies. We use the symbol “x” to denote the poles of the closed-loop SISO and the symbol circle “o” to denote the zeros of the open-loop SISO. If the pole and zero coincide then the symbol ∶ will be used to represent this situation. To study the dynamic behavior of the five genes to respond to the perturbation of silica, we consider the SISO system which takes one of the five genes as the output and silica as the input.

Changes in expression levels of genes and proteins in the regulatory networks will lead to status transition of the cells from normal cells to abnormal cells. One way to correct molecular changes is to transform cells from an undesirable state to a desirable one by altering gene or protein expressions. Now the question is whether we can use potentially therapeutic interventions to change gene or protein expressions from undesired states to desired states. This important issue can be addressed by examining the controllability of gene regulatory networks.

The fundamental controllability in gene regulation is associated with two questions. The first question is whether an input (therapy) can be found such that the system states can be driven from the undesired initial value to the desired values in a given time interval. The second question is how difficult it may be to change the system from undesired states to the desired states if the system is controllable.

The system (regulatory network) is called controllable, if for any state of the system, there exists a finite time and an admissible control function such that the system can achieve the desired state transition. In other words, the state controllability indicates that we can find an input to change the states from any initial value to any final value within some finite time. The controllability provides a binary information about whether the system is controllable or not, but it does not provide a quantitative measure to quantify the amount of control effort needed to accomplish the control task. It has been recognized that to get into insides of controllability of the system it is indispensible to define a quantity to measure how the system is controllable. In other words, we need to develop a measure to evaluate the amount of control efforts required to change the system from the initial state to the desired state

The rank of controllability matrix of the system in the state-space model of this partial TGF-β pathway under perturbation of silica in both normal and SSc fibroblasts is equal to 5 which is the number of the state variables in the model. Thus, TGF-β pathway is controllable in both normal and SSc fibroblasts. However, the conditional numbers of the controllability matrix of the system for normal and SSc fibroblasts were 80 and 398, respectively, which showed that the conditional number of the controllability matrix for the SSc fibroblasts is much larger than that for normal fibroblasts. Therefore, much more control efforts are required to change gene expressions to desired levels for the SSc fibroblasts than that for the normal fibroblasts. This implies that the controllability of this network between the normal and SSc fibroblasts are differentiable.

In the past, large efforts have been devoted to studying the function of individual genes and static properties of biological pathways. However, the molecular concentrations and activities in living organisms are in constant change as a result of their interactions. The pathogenesis of disease involves evolution and temporal process. The functions of the cells, tissues and entire organisms are not only due to the steady states of the biological pathways, but also due to the dynamic interactions of biological pathways with the external environments. Although investigation of the function of individual genes, proteins and steady-states of the biological pathways is still valuable, it is time to devote more efforts and resources to study dynamic behaviors and properties of the biological pathways. It is dynamic properties that play a central role in giving rise to the function of cells and organisms

To exemplify this principle, we studied the differential dynamic properties of a partial TGF-β signaling network under perturbation of silica between normal and SSc fibroblasts. We took this network as a dynamic system and performed dynamic system analysis. Investigation of differential dynamics of this network between the normal and SSc fibroblasts consisted of three steps. The first step was to use the EM algorithm and Kalman filter to estimate the parameters in the state-space model of this TGF-β signaling network. The second step was to study stability, the transient response and controllability of the system, and to perform root-locus analysis based on the identified state-space model of the gene network. The third step was to assess whether the dynamic properties of this network between the normal and SSc fibroblasts were different.

Our study in dynamic analysis of these gene regulations addressed several remarkable issues. First, the stability analysis may be used as a powerful tool for identifying biological pathways that are associated with the diseases. The stability is one of systems-level principles underlying complex biological pathways

Second, root-locus analysis can provide valuable information for finding genes that show strong differential dynamics between normal and abnormal cells. Not all genes in the unstable pathway show unstable dynamics. Expressions of some genes in the unstable pathway may be stable themselves. Our task is to distinguish the genes that show stable expressions from those show unstable expressions in the unstable pathway. The state transition matrix of the state-space model of the studied gene network in the SSc fibroblasts has three poles that were in the right hand sides of the complex s-plane (

Third, the controllability of biological pathway is an important property of the system. Germline or somatic mutations lead to the subsequent transcriptional and translational alterations which will finally cause diseases. Therapeutic interventions such as radiation, drug and gene therapy are intended to alter gene expressions from an undesired state to a desired or normal state. Gene regulation is a complex biological system. Theoretic and practical analyses in modern control theory demonstrate that there exist systems which we are not able (or find difficult) to change from undesired states to desired states of gene regulation. Now the question arises: are all biological pathways controllable? Are degrees of controllability of the biological pathways different between normal and abnormal Cells? The controllability measures the ability to move a system around in its entire state space using certain admissible intervention. In this report, we developed a conditional number of controllability matrix, to measure the degree of controllability of the system. Our results show that although a gene expression network responding to silica in both normal and SSc fibroblasts is controllable, the degree of controllability of this regulatory network between the normal and SSc fibroblasts is different. This regulatory network in the SSc fibroblasts has a low degree of controllability. In other words, adjustment of regulation of genes in the network by external intervention in the SSc fibroblasts is more difficult than that in the normal fibroblasts. We suspect that the degree of controllability is correlated with the severity of the diseases. When the diseases are at the initial stages, the biological systems are easy to move from abnormal states to the normal states. The degree of controllability of the system will provide valuable information on the curability of the diseases. Although in the past a number of authors have studied dynamic properties of biological networks, their studies have mainly used kinetic data or artificial data and nonlinear models

In summary, dynamic properties of the biological systems are intrinsic system properties. The gene expressions are the phenotype of the cells. Their changes are governed by the hidden dynamic properties of the gene regulatory systems. It is dynamic properties that determine the phenotypes of the cells. This report represents a paradigm shift from the studies of individual components and static properties of the system to the studies of dynamic properties of the system consisting of individual components.

Although the preliminary results are appealing, they suffer from several limitations. First, sample sizes were small. Small sample size will limit the accuracy of the state-space models for biological pathways, which in turn affect estimation of dynamic properties of the systems. No much attention in control theory has been paid to investigation of impact of uncertainty inherent in dynamic systems on dynamic properties of the system. We will treat biological networks as stochastic dynamic system and study dynamic properties of stochastic dynamic systems in the future. Second, the quantities to characterize the dynamic properties are essentially random variables. Their distributions are unknown. We have not developed statistical methods to test significant differences in the dynamic properties of the pathways between the normal and abnormal cells. Third, the relations between the dynamic properties of the genes and their genotypes have not been investigated. Fourth, we have not performed large-scale dynamic analysis of the biological pathways. More theoretical development and large-scale real data analysis for the dynamic properties of the biological pathways are urgently needed.

Skin biopsies of clinically uninvolved skin (3 mm punch) were obtained from 5 patients with SSc and 5 normal controls after informed consent was granted. All five patients fulfilled American College of Rheumatology criteria for SSc

The skin sample was transported in Dulbecco's Modified Essential Media (DMEM) with 10% fetal calf serum (FCS) (supplemented with an antibiotic and antimycotic) for processing the same day. The tissue sample was washed in 70% ethanol, PBS, and DMEM with 10% FCS. Cultured fibroblast cell strains were established by mincing tissues and placing them into 60 mm culture dishes secured by glass cover slips. The primary cultures were maintained in DMEM with 10% FCS and supplemented with antibiotic and antimycotic.

The 5th passage of fibroblast strains were plated at a density of 2.5×10^{5} cells in a 35 mm dish and grown until 80% confluence. Culture media then were replaced with FCS–free DMEM containing different doses (1, 5, 10, 25 and 50 µM) of silica particles obtained from Sigma-Aldrich, St Louis, MO. After 24-hour culture at this condition, the fibroblasts were harvested for extraction of RNA. The RNAs were examined with RT-PCR for gene expression of

Quantitative real time RT-PCR was performed using an ABI 7900 sequence detector (Applied Biosystems, Foster City, CA). The specific primers and probes for each gene were purchased through Assays-on-Demand from Applied Biosystems. As described previously (19), total RNA from each sample was extracted from cultured fibroblasts described above using TRIzol reagent (Invitrogen Life Technology) and treated with Dnase I (Ambion, Austin, TX). cDNA was synthesized using Superscript II reverse transcriptase (Invitrogen Life Technology). Synthesized cDNAs were mixed with primers/probes in the 2× Taqman universal PCR buffer, and then assayed on an ABI 7900. The data obtained from assays were analyzed with SDS 2.1 software (Applied Biosystems). The amount of total RNA in each sample was normalized with 18 S rRNA transcript levels.

A biological pathway is taken as a dynamic system. The biological system is modeled by linear state-space equations defined as_{k}_{k}_{k}

A fundamental and widely applicable parameter estimation method is Maximum Likelihood (ML) method that maximizes the likelihood of the observed data with respect to parameters. However, the state-space models involve unobserved state variables that are unavailable. It makes calculation of the likelihood in the setting of state-space models very difficult. To solve this problem, we use expectation-maximization (EM) Algorithm that is widely used iterative parameter estimation method

For the convenience of presentation, equation (2) can be rewritten as_{k}_{k}_{k}_{1}∼_{1}, _{1}). Let a sequence of input-output data samples and the state be denoted by

Calculate the expectation of the augmented log-likelihood function of both the observed data and hidden state variables defined as follows:_{θ}_{θ}_{N}_{N}_{k|k}_{k|k}_{k|k}_{−1} are calculated by the Kalman filter equations as follows:

Maximizing the likelihood function defined in equation (7) with respect to parameters yields

First we define a matrix product operation of two matrices called Hadamard product, denoted by ○, as element wise product, i.e._{mod}, as the vector in which all elements corresponding to the zeros elements in the matrix Γ are deleted. We define a modification of the matrix as the matrix in which if intersection of the row and column corresponds to the zeros elements in the matrix Γ then such row and column are deleted. The equation for estimation of parameters (18) is reduced to

Response of a biological pathway to perturbation of internal and external stimuli has two parts: the transient and the steady state response. The time varying process generated in going from the initial state to the final state in the response to the perturbation of the internal and external stimuli is called transient response. Steady-state response studies the system behavior at infinite time. Transient-response analysis of biological pathways can be used to quantify their dynamics. It can reveal how fast the biological pathways respond to perturbation of environments and how accurately the pathways can finally achieve the desired steady-state values. It can also be used to study damped vibration behavior and stability of the biological pathways.

The transient response of the dynamic systems depends on the input signals. Different signal will cause different response. There are numerous types of signal in practice. For the convenience of comparison, we consider two types of signals: (i) unit-step signal and (ii) unit-impulse signal as shown in

The transient response of dynamic systems can be studied by transfer function that is used to characterize the input-output relationships of a linear, time-invariant, differential equation system. The transfer function is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input under the assumption of zero initial conditions. The transfer function of the response of the biological pathway to unit-step and unit-impulse input signals are given by

The most important dynamic property of biological pathways is concerned with stability. Dynamic systems are called stable if their variables such as gene expressions return to, or towards, their original or equilibrium states following internal and external perturbations. For any practical purpose, the biological pathways must be stable. Unstable gene regulations will lead to the malfunction or even the death of the cells. A biological pathway will remain at steady state until occurrence of external perturbation. Depending on dynamic behavior of the system after perturbation of environments, the steady-states of the system are either stable (the system returns to the initial state or changes to other steady-states) or unstable (the system leaves the initial equilibrium state).

One of the methods for assessing the stability of the linear dynamic systems is to analyze eigenvalues of the state transition matrix

Open and close loop poles and zeros largely determine the stability and performance of the open and close systems. They provide valuable information on how to improve stability and transient response of the systems. Root-locus analysis, in which the roots of the characteristic equation of the closed-loop system are plotted for all values of a system parameter, is a powerful tool for study and design of dynamic pathway. The loop gain is often chosen to be the parameter. Varying the gain value will change the location of the closed-loop poles.

Consider a SISO system that consists of a gene regulator and an input to the gene regulator shown in

In general,

A dynamic system is called controllable, if there is an admissible control function, which can change the system from any given initial state to any finite state or to the origin of the state space in the finite time. Define the controllability matrix of the system as ^{n}^{−1}

We use the condition number of the controllability matrix to measure the degree of the controllability of the system. The condition number of the controllability matrix is defined as ^{−} is a generalized inverse of the matrix _{k}_{k}

Equation (14) can be reduced to^{−}||||