Introduction to Systems Biology (cont'd)

Definition: Most would agree that a system qualifies as complex if the overall behaviour of the system cannot be intuitively understood in terms of the individual components or interactions.

Sound familiar? Concept of emergence!

Two essential features of complex systems:

• Nonlinear interactions or processes
• Feedback loops

Definition: Positive feedback is exhibited when system components increase (excite) their own activity.

Result: Unstable divergent behaviour, but when the mechanism is constrained by saturating effects, can lead to 'locked-in' states and memory.

Examples:

• Exponential growth

Definition: Negative feedback is exhibited when system components inhibit their own activity.

Result: Stabilization (generally), but can lead to instability and oscillations if there is a time lag in the feedback.

Examples:

• Thermostat
• Logistic growth

State variables vs. Parameters

Whether an object is a state variable or a parameter depends on context and time scale.

Example: Enzyme concentrations

• Enzyme concentrations will be parameters in models of metabolism (time scale of seconds to minutes)
• Enzyme concentrations will be state variables in models of gene regulation (time scale of minutes to hours)

State variables vs. Parameters

Asymptotic vs. Transient Behavior

Definition: The long-time behavior of a system is called its asymptotic behavior.

Definition: The behavior of a system that leads from the initial state to the asymptotic behavior is called its transient behavior.

Which behavior is of interest depends on the question and the relevant time scale.

Examples of asymptotic behavior include (1) fixed points, (2) periodic behavior, (3) quasi-periodic behavior, (4) blow-up, and (4) chaos.

Linearity vs. Nonlinearity

Definition: Linearity between inputs and outputs means the output is directly proportional to the inputs.

\[ y = ax_{1} + bx_{2} \]

Definition: Nonlinearity between inputs and outputs means the the relationship is not linear.

Linearity vs. Nonlinearity

Most common nonlinear relationships in biology involve a saturating response.

Global vs. Local Behavior

Very similar to the distinction between asymptotic vs. transient behavior.

Definition: Local behavior occurs near fixed points. We can consider a linear approximation of the system at these fixed points.

“However, the global behaviour of systems is often tightly constrained by their behaviour around a handful of nominal operating points.”
- Ingalls

Deterministic vs. Stochastic Models

Definition: A mathematical model is called deterministic if its behaviour is exactly reproducible.

Definition: A stochastic model allows for randomness in its behaviour.

Generally, deterministic models are easier to work with and analyze.

More detailed model:

Different responses can be seen!