We'll start with a relatively simple system, phospho-activating enzymes. These are enzymes (proteins with catalytic ability) that work differently depending on whether one particular amino acid residue has had a phosphate group attached to it. The process is called Reversible phosphorylation, described at the KinasePhos site:
Figure 1 (from KinasePhos)
Protein phosphorylation, performed by a group of enzymes known as kinases and phosphotransferases (Enzyme Commission classification 2.7), is a post-translational modification essential to correct functioning within the cell. The post-translational modification of proteins by phosphorylation is the most abundant type of cellular regulation. It affects a multitude of cellular signal pathways, including metabolism, growth, differentiation and membrane transport. The enzymes must be sufficiently specific and act only on a defined subset of cellular targets to ensure signal fidelity. Proteins can be phosphorylated at serine, threonine and tyrosine residues.
Catalysis is defined as "the process in which the rate of a chemical reaction is either increased or decreased by means of a chemical substance known as a catalyst." In this case we'll assume it is increased. However, the reaction itself will only take place if it's energetically favorable. The rate at which it takes place depends on the relative concentrations of substrates and products, specifically how far they are from the equilibrium position. It also depends on the quantity of catalyst.
For Phosphorylation, referring to Figure 1 above, the process usually involves splitting a phosphate off of ATP (Adenosine triphosphate), the primary source of energy within the cell (it doesn't have to be ATP, GTP and other nucleotide triphosphates can also be used), and attaching it to the Hydroxyl group of an amino acid residue. This is very favorable energetically, at least until almost all of the particular target protein has been phosphorylated. Any enzyme that performs this function is called a "kinase", in this case a "protein kinase".
Similarly, for dephosphorylation, the phosphate is removed and added to the pool of inorganic phosphate. This reaction is usually not quite as favorable energetically as phosphorylation, but enough so that only a few thousandths or less of the target is left phosphorylated at equilibrium.
Now, let's start with an enzyme that only works when it's phosphorylated. We'll assume for simplicity's sake that there is only one site where this can occur. The enzyme will be activated by a kinase (another enzyme), and deactivated by a phosphatase. Or, it can be activated by several (or many) kinases, and deactivated by several (or many) phosphatases. Likewise, any specific kinase or phosphatase can act on more than one target protein (enzyme or otherwise). But what does that enzyme do? In our case, it also acts as a kinase or a phosphatase on other enzymes. Thus, enzyme A can activate enzyme B, enzyme B can activate enzyme C, and so on. At the same time, other enzymes can be de-activating them.
Remember that the rate of any catalytic reaction depends on the quantity of catalyst, as well as the relative quantities of substrate and product. The equilibrium concentration of activated (Phosphorylated) enzyme will depend on the rates of phosphorylation and dephosphorylation of all the kinases and phosphatases that affect it. This means that a network of various interacting kinases and phosphatases can drive the quantities of active enzymes in a very complex fashion. We could make an analogy with transistors: each activatable enzyme in the network corresponds to a transistor. The quantity of activated enzyme corresponds to the voltage at the output. The combination of kinase and phosphatase activity targeting it corresponds to the voltage at the input. If we hook transistors together with resistors, the specificity of kinase or phosphatase activity of any enzyme A targeting enzyme B corresponds to a resistor linking transistor A with B.
The analogy isn't exact, but in general, a cell whose cytoplasm contains 100 activatable kinase/phosphatase enzymes hooked into a network would have roughly the same computing ability as a network of 100 transistors. Consider that most small applications require only a handful of transistors, and that the number of known genes varies from 1-5,000 even in bacteria (with humans having 20-30,000). This means that even bacteria could have a network of several hundred enzymes (each coded by a gene) working together as a "brain" to control its activity. Eukaryotes usually have many more genes (5,000 an effective minimum). Not only that, but many genes in Eukaryotes have multiple transcription types, which can produce many enzymes from one gene. Given that different transcriptions often vary through inclusion or exclusion of a functional unit, this means that a single gene can "mix and match" phosphorylation sites for activation with active sites for performing activation or deactivation on other enzymes. It would be easy for even the simplest Eukaryote to have a network of several thousand different enzymes acting as its "chemical brain". Just to add more complexity, many enzymes have multiple sites of phosphorylation, often with different effects on activity.
While phospho-activation is probably the most common form of enzyme activation, there are other types of activation. Most of them work pretty much the way phospho-activation does, for our purposes.
An important thing to remember about analog computers is that each node (transistor, enzyme, etc.) can vary within a range. It can assume any value in that range, not just the "1" and "0" that digital computers use. Digital computers are generally made out of analog computers, by linking two transistors into a Flip-flop. Thus, to create a digital system out of transistors, at least two nodes capable of assuming any value within a range are linked together into a system that can only know "0" and "1". This represents a tremendous loss of intelligence relative to analog networks of transistors. Or enzymes.
While most of the activation systems that have been studied are digital, this doesn't mean most of those in the cell are. It's a lot easier to study digital systems in the cell, and those are the ones most likely to be noticed. This means that, potentially, a cell could have a surprisingly smart "chemical brain"
Next: How Smart is the Cell? Part II: The Gene Activation network as an Analog Computer