
I have noticed that man made machines are full of spinning parts such as cogs, wheels, turbines etc but as far as I know no living thing has ever evolved the wheel or anything remotely similar. Is this so and if so why?

Wheels, Cars, and Expressways
Although small to us, cells are huge spaces compared to the size of the proteins inside them: The radius of cells is 10–6–10–5 m and that of the average protein is about 10–9 m (1). That is at least 3 orders of magnitude difference (nm vs. μm), which is comparable to the scales we have to traverse in our everyday life (m vs. km). The cell, just as we do, relies on machines to carry out navigation.
The cell contains polymers, called actin cables and microtubules—the expressways—on which various protein motors walk. The function of these motors ranges from moving these filaments, providing structural support, to being the delivery trucks of the cell. Although the structure of one of the most abundant types of motors, kinesin, does not resemble a wheel, that of dynein contains a domain in its motor region that has a wheel-like structure (see http://physics.berkeley.edu/research/yildiz/research.html for a picture of dynein). This domain, named AAA (ATPases associated with cellular activities), is highly conserved and is present in other molecular machines (2). For instance, its structure has been found in a subunit of DNA polymerase III (clamploader δ subunit) and in the RuvB motor, which is involved in resolving catenated DNA in Holiday junctions.
The AAA domains use the energy of ATP hydrolysis and turn it into motion and forces. ATP hydrolysis creates changes in the conformation of the protein structure, which are then relayed and amplified onto other effector domains. It is the use of the ATP energy by the AAA domain that allows dynein to walk on microtubules or RuvB to move DNA. This wheel-like structure is the force production element of a motor. In a way, it is like the piston chamber in a car.
Some lipid vesicles (acting as transport “envelopes”) are transported by many molecular motors (e.g., kinesins and dyneins) bound to their membrane. Some of these motors interact only transiently with their track, so in a way, a vesicle is a type of rotating “wheel” making its way across the cell.
Turbines, Generators, and “Electric” Motors
The AAA domain looks like a wheel, but it doesn’t rotate. So, if you are looking for parts that resemble an actual motor or a turbine, we need to look at two other structures: the bacterial flagellar motor and the F1/F0 ATP synthase.
The bacterial motor is responsible for moving bacteria to favorable environments and getting it out of bad ones. It is the mechanical output of the bacteria’s sensory machinery. The motor looks like something you could imagine powering a human-made machine (3). A rotor complex rotates around stator subunits to move the flagellum (the long, hair-like structure that propels bacteria), using a transmembrane proton gradient as fuel. The motor can modulate its speed and rotate at up to several hundred hertz and can even reverse direction.
Another rotating mechanical element in the cell that is driven by a transmembrane proton gradient is the F1/F0 ATP synthase (4). ATP is the main energy currency in the cell, and to replenish it, under aerobic conditions, a set of reactions called “oxidative phosphorylation” drive ATP synthesis. The last step in this reaction pathway is the conversion of ATP from ADP and Pi (inorganic phosphate) by the F1/F0 synthase. The enzyme complex uses the electrochemical potential of the membrane (the proton gradient) to drive its rotation, which is required to form ATP. The enzyme can also replenish the proton gradient by using ATP hydrolysis to rotate in the opposite direction and move H+ ions into the intermembrane space. This complex can thus act both as a generator (turning motion into a readily usable energy form) and as a motor (using energy to create motion). It also looks a lot like a rotating electricity generator.
The F1/F0 complex is one of the most ubiquitous mechanical elements inside cells—it is present in bacteria, mitochondria, and chloroplasts—and is also one of the most efficient (greater than 80% efficiency in converting motion into ATP). Clearly nature likes this design!
One last structure that looks like a cartwheel is in the inside of cilia: the axoneme (this one is microtubule based). But I will let you look that one up!
References
1. Bionumbers: http://bionumbers.hms.harvard.edu/KeyNumbers.aspx
2. King, S. 2000. AAA domains and the organization of the dynein motor unit. J. Cell Sci. 113:2521–2526.
3. Berg, H. 2003. The rotary motor of bacterial flagella. 2003. Ann. Rev. Biochem. 72:19 –54.
4. Okuno, D., Iino, R., and Noji, H. 2011. Rotation and structure of F0F1 -ATP synthase. J. Biochem. 149(6):655–664.