Understanding and reducing complex systems pharmacology models based on a novel input–response index

Abstract

A growing understanding of complex processes in biology has led to large-scale mechanistic models of pharmacologically relevant processes. These models are increasingly used to study the response of the system to a given input or stimulus, e.g., after drug administration. Understanding the input–response relationship, however, is often a challenging task due to the complexity of the interactions between its constituents as well as the size of the models. An approach that quantifies the importance of the different constituents for a given input–output relationship and allows to reduce the dynamics to its essential features is therefore highly desirable. In this article, we present a novel state- and time-dependent quantity called the input–response index that quantifies the importance of state variables for a given input–response relationship at a particular time. It is based on the concept of time-bounded controllability and observability, and defined with respect to a reference dynamics. In application to the brown snake venom–fibrinogen (Fg) network, the input–response indices give insight into the coordinated action of specific coagulation factors and about those factors that contribute only little to the response. We demonstrate how the indices can be used to reduce large-scale models in a two-step procedure: (i) elimination of states whose dynamics have only minor impact on the input–response relationship, and (ii) proper lumping of the remaining (lower order) model. In application to the brown snake venom–fibrinogen network, this resulted in a reduction from 62 to 8 state variables in the first step, and a further reduction to 5 state variables in the second step. We further illustrate that the sequence, in which a recursive algorithm eliminates and/or lumps state variables, has an impact on the final reduced model. The input–response indices are particularly suited to determine an informed sequence, since they are based on the dynamics of the original system. In summary, the novel measure of importance provides a powerful tool for analysing the complex dynamics of large-scale systems and a means for very efficient model order reduction of nonlinear systems.

Appendices

Appendix 1: Differential equations for the elimination-reduced 8-state variable model of the snake venom system

The model equations for the elimination-reduced model are:

$$\begin{aligned} \begin{array}{ll} \frac{dx_{{\text {red}},{{\text{IIa}}}}}{dt} &{}=V_{{{\text{Xa:Va}}},{{\text{IIa}}}}\frac{x_{{\text {red}},{{\text{CVenom}}}}}{x_{{\text {red}},{{\text{CVenom}}}}+K_{{{\text{Xa:Va}}},{{\text{IIa}}}}}\cdot x_{{\text {env}},{{\text{II}}}}\\ &{}\quad -\left( c_{{{\text{IIa}}},{{\text{Tmod}}}} x_{{\text {env}},{{\text{Tmod}}}}+d_{{{\text{IIa}}}}\right) \cdot x_{{\text {red}},{{\text{IIa}}}},\\ \frac{dx_{{\text {red}},{{\text{Fg}}}} }{dt}&{}=p_{{{\text{Fg}}}}-\left( V_{{{\text{IIa}}},{{\text{Fg}}}}\frac{x_{{\text {red}},{{\text{IIa}}}}}{x_{{\text {red}},{{\text{IIa}}}}+K_{{{\text{IIa}}},{{\text{Fg}}}}}\right. \\ &{}\left. \quad +V_{{{\text{P}}},{{\text{Fg}}}}\frac{x_{{\text {red}},{{\text{P}}}}}{x_{{\text {red}},{{\text{P}}}}+K_{{{\text{P}}},{{\text{Fg}}}}}+d_{{{\text{Fg}}}}\right) \cdot x_{{\text {red}},{{\text{Fg}}}},\\ \frac{dx_{{\text {red}},{{\text{P}}}}}{dt}&{}=\left( V_{{{\text{IIa}}},{{\text{P}}}}\frac{x_{{\text {red}},{{\text{IIa}}}}}{x_{{\text {red}},{{\text{IIa}}}}+K_{{{\text{IIa}}},{{\text{P}}}}}\right. \\ &{} \left. \quad +V_{{{\text{APC:PS}}},{{\text{P}}}}\frac{x_{{\text {red}},{{\text{APC:PS}}}}}{x_{{\text {red}},{{\text{APC:PS}}}}+K_{{{\text{APC:PS}}},{{\text{P}}}}}\right) \\ &{}\quad \cdot x_{{\text {env}},{{\text{Pg}}}}-d_{{{\text{P}}}}\cdot x_{{\text {red}},{{\text{P}}}},\\ \frac{dx_{{\text {red}},{{\text{APC}}}}}{dt}&{}= x_{{\text {env}},{{\text{PC}}}}\cdot V_{{{\text{IIa:Tmod}}},{{\text{APC}}}}\\ &{}\quad \cdot \frac{x_{{\text {red}},{{\text{IIa:Tmod}}}}}{x_{{\text {red}},{{\text{IIa:Tmod}}}}+K_{{{\text{IIa:Tmod}}},{{\text{APC}}}}}\\ &{}\quad -\left( c_{{{\text{APC}}},{{\text{PS}}}}\cdot x_{{\text {env}},{{\text{PS}}}}+d_{{{\text{APC}}}}\right) \cdot x_{{\text {red}},{{\text{APC}}}},\\ \frac{dx_{{\text {red}},{{\text{APC:PS}}}}}{dt}&{}=c_{{{\text{APC}}},{{\text{PS}}}}\cdot x_{{\text {env}},{{\text{PS}}}}\cdot x_{{\text {red}},{{\text{APC}}}}\\ &{}\quad -d_{{{\text{APC:PS}}}}\cdot x_{{\text {red}},{{\text{APC:PS}}}},\\ \frac{dx_{{\text {red}},{{\text{IIa:Tmod}}}}}{dt}&{}=c_{{{\text{IIa}}},{{\text{Tmod}}}}\cdot x_{{\text {red}},{{\text{IIa}}}}\cdot x_{{\text {env}},{{\text{Tmod}}}}\\ &{}\quad -d_{{{\text{IIa:Tmod}}}}\cdot x_{{\text {red}},{{\text{IIa:Tmod}}}},\\ \frac{dx_{{\text {red}},{{\text{AVenom}}}}}{dt}&{}=- k_{\text {abs}} \cdot x_{{\text {red}},{{\text{AVenom}}}},\\ \frac{dx_{{\text {red}},{{\text{CVenom}}}}}{dt}&{}=k_{\text {abs}} \cdot x_{{\text {red}},{{\text{AVenom}}}} \\ &{}\quad - d_{{{\text{CVenom}}}}\cdot x_{{\text {red}},{{\text{CVenom}}}}. \end{array} \end{aligned}$$

All parameter values can be found in [2, Suppl. Fig. 2] and [17, Tables 1, 2]. The variable $x_{{\text {env}},{{\text{II}}}}$ is equal to the initial value of ${{\text{II}}}\, (x_{{{\text{II}}}}(0)$). This holds for all environmental state variables. All initial conditions can be found in [2, Suppl. Fig. 3] (Figs. 11, 12, 13, 14, 15; Tables 3, 4, 5, 6).

Fig. 11

Flowchart for the model reduction technique presented in this article

Fig. 12

Concentration–time profiles of all state variables predicted by the original model excluding those concentration–time profiles already given in Fig. 3b

Fig. 13

Comparison of concentration–time profiles for dynamical state variables of the elimination-reduced model (given in Fig. 4) based on 62-state variable model (solid) and the elimination-reduced 8-state variable model (dashed). For AVenom and CVenom the dashed and solid line coincide

Fig. 14

Comparison of fibrinogen concentration–time profile based on 62-state variable model (solid) and the model with knock-out of X and V (dot-dashed), knock-out of V (dashed) and knock-out of Pg, the inactive form of factor P (dotted). Note that the larger impact of a knock-out of V is due the fact that in this case, the impact of the activated form of factor X changes (due to the lacking complex formation of Xa and Va)

Fig. 15

Alternative elimination-reduced model of the brown snake venom–fibrinogen system. Model reduction based on randomly chosen ranking of state variables (based on run no. 2). Shown are the 12 dynamical state variables. The environmental state variables (indicated by ‘*’) are II, V, VIII, X and XI

Table 3 Ordering of the state variables based on the input–response indices for the brown snake venom–fibrinogen system up to 1 h. State variables not included in the table had an input–response index of zero

Table 4 Relative error for the 8-state variable reduced model and the 12-state variable reduced model on the interval [0, 40 h]. Given is the squared relative error for the output fibrinogen based on the 62-state variable and the reduced models (Figs. 4, 15) for different time intervals of interest (pre nadir, post nadir and for the whole interval)

Table 5 Order of the elimination-reduced model obtained by varying the user-defined error tolerance for the brown snake venom–fibrinogen system for the first hour after envenomation. Given are the dynamical state variables and the environmental state variables for each of the obtained elimination-reduced model for the different user-defined error tolerance

Table 6 Parameter value of the elimination-reduced and lumped 5-state variable model of the brown snake venom–fibrinogen system. Explanation of notation (by examples): $V_{{{\text{IIa}}},{{\text{P}}}}$ represents maximal reaction velocity and $K_{{{\text{IIa}}},{{\text{P}}}}$ the concentration of IIa were half the maximal velocity is obtained (both are Michaelis–Menten constants). In both cases, ${{\text{P}}}$ represents the factor which is being activated by this reaction. A degradation rate constants is denoted by $d_{{{\text{P}}}}$ and production rate constants by $p_{{{\text{IIa}}}}.$ A constant $c_{{{\text{IIa}}},{{\text{Tmod}}}}$ denotes a reaction rate constant for complex formation

Appendix 2: Differential equations for the elimination-reduced 7-state variable model and 13-state variable model of the PT test

The model equations for the seven-state variable elimination-reduced model for the high TF are:

$$\begin{aligned} \begin{array}{ll} \frac{dx_{{\text {red}},{{\text{IIa}}}}}{dt} &{}=V_{{{\text{Xa}}},{{\text{IIa}}}}\frac{x_{{\text {red}},{{\text{Xa}}}}}{x_{{\text {red}},{{\text{Xa}}}}+K_{{{\text{Xa}}},{{\text{IIa}}}}}\cdot x_{{\text {env}},{{\text{II}}}}-d_{{{\text{IIa}}}}\cdot x_{{\text {red}},{{\text{IIa}}}},\\ \frac{dx_{{\text {red}},{{\text{Fg}}}} }{dt}&{}=-V_{{{\text{IIa}}},{{\text{Fg}}}}\frac{x_{{\text {red}},{{\text{IIa}}}}}{x_{{\text {red}},{{\text{IIa}}}}+K_{{{\text{IIa}}},{{\text{Fg}}}}}\cdot x_{{\text {red}},{{\text{Fg}}}}-d_{{{\text{Fg}}}}\cdot x_{{\text {red}},{{\text{Fg}}}},\\ \frac{dx_{{\text {red}},{{\text{F}}}} }{dt}&{}=V_{{{\text{IIa}}},{{\text{Fg}}}}\frac{x_{{\text {red}},{{\text{IIa}}}}}{x_{{\text {red}},{{\text{IIa}}}}+K_{{{\text{IIa}}},{{\text{Fg}}}}}\cdot x_{{\text {red}},{{\text{Fg}}}}-d_{{{\text{F}}}}\cdot x_{{\text {red}},{{\text{F}}}},\\ \frac{dx_{{\text {red}},{{\text{Xa}}}} }{dt}&{}=V_{{{\text{VIIa:TF}}},{{\text{X}}}}\frac{x_{{\text {red}},{{\text{VII:TF}}}}}{x_{{\text {red}},{{\text{VII:TF}}}}+K_{{{\text{VII:TF}}},{{\text{X}}}}}\cdot x_{{\text {env}},{{\text{X}}}} - c_{{{\text{TFPI}}},{{\text{Xa}}}}\cdot x_{{\text {env}},{{\text{TFPI}}}}\cdot x_{{\text {red}},{{\text{Xa}}}}\\ {} &{}\quad -d_{{{\text{Xa}}}}\cdot x_{{\text {red}},{{\text{Xa}}}},\\ \frac{dx_{{\text {red}},{{\text{VII:TF}}}} }{dt}&{}=c_{{{\text{VII}}},{{\text{TF}}}}\cdot x_{{\text {red}},{{\text{TF}}}}\cdot x_{{\text {env}},{{\text{VII}}}}-\left( V_{{{\text{Xa}}},{{\text{VII:TF}}}}\frac{x_{{\text {red}},{{\text{Xa}}}}}{x_{{\text {red}},{{\text{Xa}}}}+K_{{{\text{Xa}}},{{\text{VII:TF}}}}}\right. \\ &{}\quad \left. +V_{{{\text{TF}}},{{\text{VII:TF}}}}\frac{x_{{\text {red}},{{\text{TF}}}}}{x_{{\text {red}},{{\text{TF}}}}+K_{{{\text{TF}}},{{\text{VII:TF}}}}}\right) \cdot x_{{\text {red}},{{\text{VII:TF}}}}-d_{{{\text{VII:TF}}}}\cdot x_{{\text {red}},{{\text{VII:TF}}}},\\ \frac{dx_{{\text {red}},{{\text{VIIa:TF}}}} }{dt}&{}=\left( V_{{{\text{Xa}}},{{\text{VII:TF}}}}\frac{x_{{\text {red}},{{\text{Xa}}}}}{x_{{\text {red}},{{\text{Xa}}}}+K_{{{\text{Xa}}},{{\text{VII:TF}}}}} +V_{{{\text{TF}}},{{\text{VII:TF}}}}\frac{x_{{\text {red}},{{\text{TF}}}}}{x_{{\text {red}},{{\text{TF}}}}+K_{{{\text{TF}}},{{\text{VII:TF}}}}}\right) \cdot x_{{\text {red}},{{\text{VII:TF}}}}\\ {} &{}\quad -d_{{{\text{VIIa:TF}}}}\cdot x_{{\text {red}},{{\text{VIIa:TF}}}},\\ \frac{dx_{{\text {red}},{{\text{TF}}}}}{dt}&{}=-c_{{{\text{VII}}},{{\text{TF}}}}\cdot x_{{\text {red}},{{\text{TF}}}}\cdot x_{{\text {env}},{{\text{VII}}}}-d_{{{\text{TF}}}}\cdot x_{{\text {red}},{{\text{TF}}}}. \end{array} \end{aligned}$$

All parameter values and initial conditions for the state variables can be found in [2, Suppl. Figs. 2, 3]. The state variable $x_{{\text {env}},{{\text{II}}}}$ is equal to the initial value of ${{\text{II}}}\, (x_{{{\text{II}}}}(0)$). This holds for all environmental state variables.

The model equations for the 13-state variable elimination-reduced model for the low TF are:

$$\begin{aligned} \begin{array}{ll} \frac{dx_{{\text {red}},{{\text{IIa}}}}}{dt} &{}=V_{{{\text{Xa}}},{{\text{IIa}}}}\frac{x_{{\text {red}},{{\text{Xa}}}}}{x_{{\text {red}},{{\text{Xa}}}}+K_{{{\text{Xa}}},{{\text{IIa}}}}}\cdot x_{{\text {env}},{{\text{II}}}}-d_{{{\text{IIa}}}}\cdot x_{{\text {red}},{{\text{IIa}}}},\\ \frac{dx_{{\text {red}},{{\text{Fg}}}} }{dt}&{}=-V_{{{\text{IIa}}},{{\text{Fg}}}}\frac{x_{{\text {red}},{{\text{IIa}}}}}{x_{{\text {red}},{{\text{IIa}}}}+K_{{{\text{IIa}}},{{\text{Fg}}}}}\cdot x_{{\text {red}},{{\text{Fg}}}}-d_{{{\text{Fg}}}}\cdot x_{{\text {red}},{{\text{Fg}}}},\\ \frac{dx_{{\text {red}},{{\text{F}}}} }{dt}&{}=V_{{{\text{IIa}}},{{\text{Fg}}}}\frac{x_{{\text {red}},{{\text{IIa}}}}}{x_{{\text {red}},{{\text{IIa}}}}+K_{{{\text{IIa}}},{{\text{Fg}}}}}\cdot x_{{\text {env}},{{\text{Fg}}}}-d_{{{\text{F}}}}\cdot x_{{\text {red}},{{\text{F}}}},\\ \frac{dx_{{\text {red}},{{\text{Xa}}}} }{dt}&{}=\left( V_{{{\text{VIIa:TF}}},{{\text{X}}}}\frac{x_{{\text {red}},{{\text{VII:TF}}}}}{x_{{\text {red}},{{\text{VII:TF}}}}+K_{{{\text{VII:TF}}},{{\text{X}}}}}+V_{{{\text{IXa}}},{{\text{X}}}}\frac{x_{{\text {red}},{{\text{IXa}}}}}{x_{{\text {red}},{{\text{IXa}}}}+K_{{{\text{IXa}}},{{\text{X}}}}}\right. \\ &{}\quad \left. +V_{{{\text{IXa:VIIIa}}},{{\text{X}}}}\frac{x_{{\text {red}},{{\text{IXa:VIIIa}}}}}{x_{{\text {red}},{{\text{IXa:VIIIa}}}}+K_{{{\text{IXa:VIIIa}}},{{\text{X}}}}}\right) \cdot x_{{\text {env}},{{\text{X}}}} - c_{{{\text{TFPI}}},{{\text{Xa}}}}\cdot x_{{\text {env}},{{\text{TFPI}}}}\\ {} &{}\quad \cdot x_{{\text {red}},{{\text{Xa}}}} -d_{{{\text{Xa}}}}\cdot x_{{\text {red}},{{\text{Xa}}}},\\ \frac{dx_{{\text {red}},{{\text{VII:TF}}}} }{dt}&{}=c_{{{\text{VII}}},{{\text{TF}}}}\cdot x_{{\text {red}},{{\text{TF}}}}\cdot x_{{\text {env}},{{\text{VII}}}}-\left( V_{{{\text{Xa}}},{{\text{VII:TF}}}}\frac{x_{{\text {red}},{{\text{Xa}}}}}{x_{{\text {red}},{{\text{Xa}}}}+K_{{{\text{Xa}}},{{\text{VII:TF}}}}} \right. \\ &{}\quad \left. +V_{{{\text{TF}}},{{\text{VII:TF}}}}\frac{x_{{\text {red}},{{\text{TF}}}}}{x_{{\text {red}},{{\text{TF}}}}+K_{{{\text{TF}}},{{\text{VII:TF}}}}}\right) \cdot x_{{\text {red}},{{\text{VII:TF}}}}-d_{{{\text{VII:TF}}}}\cdot x_{{\text {red}},{{\text{VII:TF}}}},\\ \frac{dx_{{\text {red}},{{\text{VIIa:TF}}}} }{dt}&{}=\left( V_{{{\text{Xa}}},{{\text{VII:TF}}}}\frac{x_{{\text {red}},{{\text{Xa}}}}}{x_{{\text {red}},{{\text{Xa}}}}+K_{{{\text{Xa}}},{{\text{VII:TF}}}}}+V_{{{\text{TF}}},{{\text{VII:TF}}}}\frac{x_{{\text {red}},{{\text{TF}}}}}{x_{{\text {red}},{{\text{TF}}}}+K_{{{\text{TF}}},{{\text{VII:TF}}}}}\right) \cdot x_{{\text {red}},{{\text{VII:TF}}}}\\ {} &{}\quad -d_{{{\text{VIIa:TF}}}}\cdot x_{{\text {red}},{{\text{VIIa:TF}}}},\\ \frac{dx_{{\text {red}},{{\text{TF}}}} }{dt}&{}=-c_{{{\text{VII}}},{{\text{TF}}}}\cdot x_{{\text {red}},{{\text{TF}}}}\cdot x_{{\text {env}},{{\text{VII}}}}-d_{{{\text{TF}}}}\cdot x_{{\text {red}},{{\text{TF}}}},\\ \frac{dx_{{\text {red}},{{\text{VIII}}}}}{dt} &{}=-V_{{{\text{IIa}}},{{\text{VIII}}}}\frac{x_{{\text {red}},{{\text{IIa}}}}}{x_{{\text {red}},{{\text{IIa}}}}+K_{{{\text{IIa}}},{{\text{VIII}}}}}\cdot x_{{\text {red}},{{\text{VIII}}}}-d_{{{\text{VIII}}}}\cdot x_{{\text {red}},{{\text{VIII}}}},\\ \frac{dx_{{\text {red}},{{\text{VIIIa}}}}}{dt} &{}=V_{{{\text{IIa}}},{{\text{VIII}}}}\frac{x_{{\text {red}},{{\text{IIa}}}}}{x_{{\text {red}},{{\text{IIa}}}}+K_{{{\text{IIa}}},{{\text{VIII}}}}}\cdot x_{{\text {red}},{{\text{VIII}}}}-\left( c_{{{\text{IXa}}},{{\text{VIIIa}}}}\cdot x_{{\text {red}},{{\text{IXa}}}}+d_{{{\text{VIIIa}}}}\right) \\ {} &{}\quad \cdot x_{{\text {red}},{{\text{VIIIa}}}},\\ \frac{dx_{{\text {red}},{{\text{IXa}}}}}{dt} &{}=V_{{{\text{VIIa:TF}}},{{\text{IX}}}}\frac{x_{{\text {red}},{{\text{VIIa:TF}}}}}{x_{{\text {red}},{{\text{VIIa:TF}}}}+K_{{{\text{VIIa:TF}}},{{\text{IX}}}}}\cdot x_{{\text {env}},{{\text{IX}}}}-\left( c_{{{\text{IXa}}},{{\text{VIIIa}}}}\cdot x_{{\text {red}},{{\text{VIIIa}}}}+d_{{{\text{IXa}}}}\right) \\ {} &{}\quad \cdot x_{{\text {red}},{{\text{IXa}}}},\\ \frac{dx_{{\text {red}},{{\text{IXa:VIIIa}}}}}{dt} &{}=c_{{{\text{IXa}}},{{\text{VIIIa}}}}\cdot x_{{\text {red}},{{\text{IXa}}}}\cdot x_{{\text {red}},{{\text{VIIIa}}}}-d_{{{\text{IXa:VIIIa}}}}\cdot x_{{\text {red}},{{\text{IXa:VIIIa}}}},\\ \frac{dx_{{\text {red}},{{\text{Va}}}}}{dt} &{}=V_{{{\text{IIa}}},{{\text{V}}}}\frac{x_{{\text {red}},{{\text{IIa}}}}}{x_{{\text {red}},{{\text{IIa}}}}+K_{{{\text{IIa}}},{{\text{V}}}}}\cdot x_{{\text {env}},{{\text{V}}}}-d_{{{\text{IXa:VIIIa}}}}\cdot x_{{\text {red}},{{\text{Va}}}},\\ \frac{dx_{{\text {red}},{{\text{Xa:Va}}}}}{dt} &{}=c_{{{\text{Xa}}},{{\text{Va}}}}\cdot x_{{\text {red}},{{\text{Xa}}}}\cdot x_{{\text {red}},{{\text{Va}}}}-d_{{{\text{Xa:Va}}}}\cdot x_{{\text {red}},{{\text{Xa:Va}}}}. \end{array} \end{aligned}$$

All parameter values and initial conditions for the state variables can be found in [2, Suppl. Figs. 2,  3]. The state variable $x_{{\text {env}},{{\text{II}}}}$ is equal to the initial value of ${{\text{II}}}\, (x_{{{\text{II}}}}(0)$). This holds for all environmental state variables.