Task-oriented modular decomposition of biological networks: trigger mechanism in blood coagulation

Abstract

Analysis of complex time-dependent biological networks is an important challenge in the current postgenomic era. We propose a middle-out approach for decomposition and analysis of complex time-dependent biological networks based on: 1), creation of a detailed mechanism-driven mathematical model of the network; 2), network response decomposition into several physiologically relevant subtasks; and 3), subsequent decomposition of the model, with the help of task-oriented necessity and sensitivity analysis into several modules that each control a single specific subtask, which is followed by further simplification employing temporal hierarchy reduction. The technique is tested and illustrated by studying blood coagulation. Five subtasks (threshold, triggering, control by blood flow velocity, spatial propagation, and localization), together with responsible modules, can be identified for the coagulation network. We show that the task of coagulation triggering is completely regulated by a two-step pathway containing a single positive feedback of factor V activation by thrombin. These theoretical predictions are experimentally confirmed by studies of fibrin generation in normal, factor V-, and factor VIII-deficient plasmas. The function of the factor V-dependent feedback is to minimize temporal and parametrical intervals of fibrin clot instability. We speculate that this pathway serves to lessen possibility of fibrin clot disruption by flow and subsequent thromboembolism.

Figure 1

Kinetics of blood clotting in a homogeneous system. (A) Fibrin generation in plasma activated with 5 pM TF, computer simulations. using the model defined by Eqs. S1–S35 in the Supporting Material. Parameters in the panel: clot time (i.e., the time required to achieve 50% fibrinogen cleavage); approximate fibrin jellification threshold; and total fibrin concentration at the end of experiment. (B) Optical density increase in plasma upon addition of 5 pM TF, three independent experiments each performed with pools from three normal plasmas.

Figure 2

Model reduction. Kinetics of coagulation in plasma activated with TF at 0.01 pM: (A) the VIIa-TF complex; (B) factor Xa; (C) thrombin; (D) fibrin. Model reduction steps: 1), No reduction, computer simulations with the original model; 2), Step A, nonessential components are removed; and 3), Step B, temporal decomposition.

Figure 3

Dynamics of the system. (A and B) Projections of phase space of the complete model (see Eqs. S1–S35 in the Supporting Material) on the x_3p-x₂ plane in linear (A) or semilogarithmic (B) scale. Null-cline projections dx₂/dt = 0 and dx_3p/dt = 0 are thick solid and shaded lines, respectively. Arrows show directions of phase trajectories. Phase trajectories are thin solid lines; thin dotted line in panel B shows trajectories at high TF concentrations, when limitation on the amount of inactive precursors becomes important. (C) Phase portrait for the reduced robusterized model (see expressions in Eq. 3).

Figure 4

The effect of factor V on the time course of clotting. Theoretical (A, B, D, and E) and typical experimental (C and F) kinetics of clotting in normal (A–C) and factor V-deficient (D–F) plasma. (A and D) Computer simulations using the original model; (B and E) reduced model (Eqs. 2); (C and F) experiments. TF concentrations are: (A and B) 0.64, 0.32, 0.16, 0.08, 0.04, 0.02, 0.01, and 0 pM, top to bottom; (D and E) 64, 32, 16, 8, 4, 2, 1, and 0 pM, top to bottom; and (C and F) shown in the figure.

Figure 5

The effect of factor V on the stimulus-response relationship. Theoretical (A and B) and experimental (C and D) dependences of final fibrin clot density (after 5 h) on TF concentration in either normal (A and C) or factor V-deficient (B and D) plasmas. (A and B, solid curves) Computer simulations with the original model. (Dotted curves) Calculations with reduced models. (B, inset) Compares the curves for complete factor V deficiency (0% of protein activity) and 1% deficiency. (C) Results presented are means ± SE of n = 2 separate experiments each performed in quadruplicate using pools from three normal plasmas. (D) Results are means ± SE for n = 3 experiments.

Figure 6

Constructing a trigger: temporal and terminal responses of different enzyme cascades. The diagram illustrates the effect of cascade structure on their response. The cascades (A–D) begin with an initial signal (enzyme E₁), which is a decaying exponent. On the right, two system responses are shown: the dependence of the product P concentration on time (temporal response), and the dependence of the final (achieved at infinite time) concentration of P on the amplitude of the initial signal (terminal response). In a simple one-step cascade (A), both dependences are linear near zero. Addition of steps in a cascade (B) can make the temporal response nonlinear, with no effect on the terminal response. If the added step is a rapid variable (C), even this advantage is lost. However, a positive feedback (D) can make both responses nonlinear and create a true trigger. This latter case corresponds to blood coagulation if E₁ is factor Xa, E₂ is thrombin, and P is fibrin.

Figure 7

Modular decomposition of blood coagulation. Findings of this study combined with those from other reports allow us to identify six modules responsible for different subtasks. The reduced models (Eqs. 1 and 2) of this study include modules 1, 2, and 3.