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Gregory T. Reeves and Scott E. Fraser are at the Biological Imaging Center, Beckman Institute, Division of Biology, California Institute of Technology, Pasadena, California, United States of America.

The robust design of complex systems, such as cruise control, requires a careful balance of several objectives. As biological systems are no different, an engineering approach to these systems proves useful.

Many dynamical systems can be sufficiently described by a set of ordinary differential equations (ODEs), which model how the variables change in time (but not space). A variable, such as enzyme concentration, could be affected by three factors: rates of production, degradation/reaction, and influx/efflux. Adding these influences together will tell us how that variable evolves.

Consider a simple example of an enzyme with intracellular concentration, _{p}(_{d}(

The first step in analysis is to determine the steady-state solution (the enzyme concentration that allows for the balance between the rates of production and degradation; see example in

(A) Reaction schematic of an enzyme, E, catalyzing the conversion of substrate, S, to product, P. The final product acts as a catabolite to promote the expression of enzyme.

(B) ODE describing the change in time of enzyme concentration, _{p}(_{d}(

(C) Graphical representation of _{p}(_{d}(

To make the example concrete, consider the case in which the enzyme concentration promotes its own production (_{m} is analogous to a Michaelis-Menten constant, and assume enzyme concentration is diminished in a first-order process (see _{m}, where μ is the decay rate constant. This can be seen graphically in _{p}(_{d}(

In recent years, much research has focused on the robustness of biological systems [

In applied sciences, “sensitivity” (the opposite of robustness) is measured by a quantity called the

But what does this number mean? How do you know whether a value of the sensitivity coefficient is “good?” In the absence of further knowledge, we would like the absolute value of the sensitivity coefficient to be less than 1. This implies that any fractional change in the input will correspond to a smaller fractional change in the output. Conversely, if the absolute value is greater than 1, small fluctuations in the input will be amplified in the output.

To illustrate an application of this analysis, we return to our example in _{m} = 1 M, and μ = 0.1 s^{–1}, it is easy to calculate

To make matters worse, it is easy to show in this illustration that no feasible values of the three parameters can give a sensitivity coefficient of magnitude less than 1. In general, it may be difficult to find parameter values that ensure robust outputs. In engineered systems, this is often solved by feedback regulation.

The engineering and applied mathematics subfield of “control theory” refers to the use of feedback loops to ensure that system outputs, such as product purity, are maintained at set values (see general control loop in

(A) Schematic of a simple control loop. The process output is monitored by a sensor, and the value of this output signal is passed to a device called a controller. The controller calculates the difference between the output signal and the set point (the desired value of the output), and responds accordingly, often by physically manipulating an input parameter, such as a control valve.

(B) Schematic of cruise control. The car is the process, and the car's speed is the output. A speedometer sensor within the car tells the cruise control the car's speed. The actuator on the cruise control then responds by opening or closing the throttle, allowing air intake into the engine.

Use of feedback control always comes with a cost, or design trade-off: an increase in robustness/performance in one area must be accompanied by a decrease somewhere else [

At this point, it is instructive to stop and consider the analogy between biological tissues and engineered systems (such as a car). Many biologists have remarked on the apparent design of biological systems, arguing that this is a false analogy. However, evolutionary theory would predict apparent design and purpose in biological systems. Therefore, regardless of the origin of this apparent design, the analogy is, at the very least, pragmatic [

To demonstrate, let us return our simple example in _{d}(^{2}. In a sense, this is a type of negative feedback in which the enzyme is responsible for its own dilution. Under these conditions, the sensitivity coefficients for all of the parameters will always have a magnitude less than 1! However, as a tradeoff, the system will generally take a longer time to reach steady state. So which performance objective is more important to the cell: robustness with respect to parameter variation, or rapid approach to steady state?

In other words, when optimizing a large system with several performance objectives, any one objective, taken in isolation, is simple to optimize. However, optimization of the whole system—that is, balancing multiple performance objectives—can be extremely difficult to achieve, especially because performance objectives are often at odds with each other. Regarding robustness, in most cases (outside of simple examples with one variable and three parameters), it is not feasible to design a system in which all sensitivity coefficients have a magnitude of less than one; more information is needed to ascertain the tolerable values of the sensitivity coefficients. What levels of fluctuations in the inputs are we expected to face? What kind of error in the output variable(s) are we prepared to tolerate? It would be a poor performance objective to be robust to variables that are either well-controlled or have narrow windows of natural variation, or to minimize the sensitivity of an output that has no need of tight control.

Furthermore, robustness to some variables may have disastrous consequences. Consider the example of cruise control, discussed earlier. If we focus on the controller (rather than the car) as the system, then the input is the speed of the car, and the output is the signal that opens or closes the throttle. If the controller itself were resistant to changes in speed of the car, the control system would be useless! In other words, determining performance objectives is a first major step in understanding the function of a biological system.

In this issue of

The authors begin with a general analysis of unbranched cell lineages, using a set of ODEs to model the population of each cell stage. In the case with no feedback regulation in the olfactory neuron cell lineage, they show that the system can only realize a steady state number of ORNs under very special conditions (i.e., parameter values). A sensitivity analysis shows that even small deviations from these conditions have drastic consequences on the steady state.

In further analysis, Lander et al. focus on several experimentally observed performance objectives—including rapid regeneration after injury, low progenitor load (stem plus precursor cells make up less than 10% of the OE), and robustness of the steady state—and ask whether feedback loops could be designed to simultaneously meet these objectives. Briefly, they find that GDF11 must act not only to slow the cell cycle of precursor cells, but also to increase the likelihood that a dividing precursor cell may produce ORNs. In a similar manner, activin must also act on the stem cell population.

Next, the authors investigated the ability of the OE to sense varying concentrations of GDF11 and activin. In this case, the performance objective is

Remarkably, the work done by Lander et al. [

Many biologists have now begun to advocate mathematical modeling, noting that “a cartoon model” is no longer sufficient [

This is the case with the study by Lander et al. The authors begin with very general arguments from physics and mathematics, which correctly describe the overall behavior of a cell lineage without feedback, and have no need to model further detail. Indeed, a more detailed model would necessarily behave in a qualitatively similar fashion, but would muddy the water on the conclusions. Noting that the simple model fails to display robustness (a critical performance objective), they add complexity step-wise until this performance objective is met. Their model is sufficient to show that feedback is vital to the stability and robustness of any such lineage. While few would be surprised at this, the authors also show what type of feedback is necessary to achieve the stated performance objectives of rapid regeneration.

Some have argued that a mathematical model is of no use unless it is able to make further predictions about the system. To do this, it is preferable to include mechanistic detail in the model [

The authors would like to thank Tuomas Brock for helpful suggestions on the manuscript.

ordinary differential equation

olfactory epithelium

olfactory receptor neuron