Chemostat principle — For organisms growing at steady state in a chemostat, how is the specific growth rate determined?

Introduction:
A chemostat maintains a microbial culture at a defined physiological state by continuous feeding and withdrawal. The central steady-state relationship links the organism’s specific growth rate to the hydraulic operating condition. This question tests the core identity mu = D.

Given Data / Assumptions:

• Perfect mixing; constant volume V and flow F.
• Single limiting nutrient; stable steady state (no washout).
• Biomass concentration and growth rate are time-invariant at steady state.

Concept / Approach:
At steady state, biomass accumulation is zero. The mass balance for cells is dX/dt = (mu − D) * X = 0, giving mu = D provided X > 0. Thus the specific growth rate is set by, and equals, the dilution rate D = F/V. It does not generally equal mu_max unless D is chosen near that value without causing washout.

Step-by-Step Solution:
Write biomass balance: dX/dt = mu * X − D * X.Impose steady state (dX/dt = 0) with nonzero X.Solve: mu = D.Interpretation: measuring or setting D determines mu at steady state.

Verification / Alternative check:
Independent estimates of mu from substrate uptake or CO2 evolution agree with D under steady conditions; when D exceeds mu_max the culture washes out, confirming the relationship’s limit.

Why Other Options Are Wrong:

• (a) False; mu is directly controlled by D.
• (c) False; mu equals mu_max only at the brink of washout.
• (d) Incorrect given (b) is true.
• (e) Not a general identity; mu depends on D and kinetics, not simply S_feed / tau.

Common Pitfalls:
Confusing instantaneous mu with mu_max; overlooking transient behavior before steady state; neglecting cell recycle systems where the simple balance changes.

Final Answer:
It equals the dilution rate and can be determined from it.