, the volumetric flow rate associated with a pressure drop isand the corresponding power consumed isSaid pipe contains volume . If the maintenance power density is , then the total power consumed (from both flow and upkeep) isMinimizing this quantity depends on precisely which variables the organism is free to manipulate, but the minimum invariably occurs when the two terms are proportional to each other. In that minimal case, the proportionality determines a relationship between and . Canceling common factors and taking a square root, That is, when using as little energy as possible, the mass flowing through the pipe must be proportional to the cube of the pipe's radius. Since flow is leakless, the total flow rate into a junction must be the total flow rate out:Substituting () then gives Murray's law with . Exponents for other networks If the network does not rely on transported material getting "swept up in the flow", but instead expects it to passively diffuse, then resistance to transport is minimized when : that is, The same law would apply to a direct-current electrical grid composed of wires of only one material, but varying diameter. For turbulent flow, transport resistance is minimized when ; that is: In general, networks intermediate between diffusion and laminar flow are expected to have characteristic exponents between 2 and 3, at least approximately. Experimental tests Murray's law has been verified in chicks; dog intestines and lungs; cat mesentery; and human intestines and lung capillaries. Mice genetically engineered to lack the blood-vessel-wall protein elastin have smaller and thinner blood vessels, but still obey Murray's law. In humans, large vessels, such as the aorta or trachea, do not appear to obey Murray's law, instead obeying a Murray's law with exponent close to 2. But flow in those vessels is also partially turbulent, and so should exhibit an exponent nearer to than to 3.Insects do not have a fully-fledged circulatory system, instead relying on passive diffusion through the haemocoel. For those networks, Murray's law predicts constant cross-sectional area, which is observed. The same arguments that imply Murray's law also imply that the distribution of tubules should exhibit a specific power law scaling with size. Plant xylem is known to exhibit that scaling except in scenarios where the passages double as structural supports. History The first phenomenon now recognized as Murray's law is Young's rule for circulatory systems, which states that two identical