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these microscopic components need to be serviced in a relatively “democratic” and efficient fashion in order to supply metabolic substrates, remove waste products, and regulate activity.
Natural selection has solved this challenge in perhaps the simplest possible way by evolving hierarchical branching networks that distribute energy and materials between macroscopic reservoirs and microscopic sites.
it was natural to conjecture that the key to the origin of quarter-power allometric scaling laws, and consequently to the generic coarse-grained behavior of biological systems, lay in the universal physical and mathematical properties of these networks.
Macroecology has whimsically been referred to as “seeing the forest for the trees.”
Revolutionary ideas like the realization that the nature of matter itself is fundamentally probabilistic, as embodied in Heisenberg’s uncertainty principle, and that space and time are not fixed and absolute, arose out of addressing the limitations of classical Newtonian thinking.
Quantum mechanics is the foundational theoretical framework for understanding materials and plays a seminal role in much of the high-tech machinery and equipment that we use. In particular, it stimulated the invention of the laser,
Similarly, relativity together with quantum mechanics spawned atomic and nuclear bombs,
A major challenge in constructing theories and models is to identify the important quantities that capture the essential dynamics at each organizational level of a system.
what was irrelevant at one scale can become dominant at another. The challenge at every level of observation is to abstract the important variables that determine the dominant behavior of the system.
“toy model.” The strategy is to simplify a complicated system by abstracting its essential components, represented by a small number of dominant variables, from which its leading behavior can be determined.
A concept related to the idea of a toy model is that of a “zeroth order” approximation of a theory, in which simplifying assumptions are similarly made in order to give a rough approximation of the exact result.
each succeeding “order” represents a refinement, an improved approximation, or a finer resolution that converges to the exact result based on more detailed investigation and analysis.
there must be a common set of network properties that transcends whether they are constructed of tubes as in mammalian circulatory systems, fibers as in plants and trees, or diffusive pathways as in cells.
the concept of space filling is simple and intuitive. Roughly speaking, it means that the tentacles of the network have to extend everywhere throughout the system that it is serving,
Space filling is simply the statement that the capillaries, which are the terminal units or last branch of the network, have to service every cell in our body so as to efficiently supply each of them with sufficient blood and oxygen.
The pipe that connects your house to the water line in the street and the electrical line that connects it to the main cable are analogs of capillaries, while your house can be thought of as an analog to cells.
Similarly, all employees of a company, viewed as terminal units, have to be supplied by resources (wages, for example) and information through multiple networks connecting them with the CEO and the management.
the terminal units of a given network design, such as the capillaries of the circulatory system that we just discussed, all have approximately the same size and characteristics regardless of the size of the organism.
As individuals grow from newborn to adult, or as new species of varying sizes evolve, terminal units do not get reinvented nor are they significantly reconfigured or rescaled.
This invariance of terminal units can be understood in the context of the parsimonious nature of natural selection.
From this perspective, the difference between taxonomic groups, that is, between mammals, plants, and fish, for example, is characterized by different properties of the terminal units of their various corresponding networks.
the continuous multiple feedback and fine-tuning mechanisms implicit in the ongoing processes of natural selection and which have been playing out over enormous periods of time have led to the network performance being “optimized.”
All the laws of physics can be derived from the principle of least action which, roughly speaking, states that, of all the possible configurations that a system can have or that it can follow as it evolves in time, the one that is physically realized is the one that minimizes its action.
we showed how Kleiber’s law for metabolic rates and, indeed, quarter-power scaling in general arises from the dynamics and geometry of optimized space-filling branching networks.
Oxygen molecules bind to the iron-rich hemoglobin in blood cells, which act as the carriers of oxygen. It is this oxidation process that is responsible for our blood being red in much the same way that iron turns red when it oxidizes to rust in the atmosphere.
blood flow rates, respiratory rates, and metabolic rates are all proportional to one another and related by simple linear relationships. Thus, hearts beat approximately four times for each breath that is inhaled, regardless of the size of the mammal.
The rate at which you use energy to pump blood through the vasculature of your circulatory system is called your cardiac power output.
It’s much harder to push fluid through a narrow tube than a wider one, so almost all of the energy that your heart expends is used to push blood through the tiniest vessels at the end of the network.
One of the basic postulates of our theory is that the network configuration has evolved to minimize cardiac output, that is, the energy needed to pump blood through the system.
When blood leaves the heart, it travels down through the aorta in a wave motion that is generated by the beating of the heart. The frequency of this wave is synchronous with your heart rate, which is about sixty beats a minute.
A generic feature of waves is that they suffer reflections when they meet a barrier,
These reflections have potentially very bad consequences because they mean that your heart is effectively pumping against itself.
To avoid this potential problem and minimize the work our hearts have to do, the geometry of our circulatory systems has evolved so that there are no reflections at any branch point throughout the network.
the theory predicts that there will be no reflections at any branch point if the sum of the cross-sectional areas of the daughter tubes leaving the branch point is the same as the cross-sectional area of the parent tube coming into it.
This so-called area-preserving branching is, indeed, how our circulatory system is constructed, as has been confirmed by detailed measurements across many mammals—and many plants and trees.
the condition of nonreflectivity of waves at branch points in pulsatile networks is essentially identical to how national power grids are designed for the efficient transmission of electricity over long distances. This condition of nonreflectivity is called impedance matching.
Without the gel, the impedance mismatch in ultrasound detection would result in almost all of the energy being reflected back from the skin,
The term impedance matching can be a very useful metaphor for connoting important aspects of social interactions.
“reflected,” such as when one side is not listening, it cannot be faithfully or efficiently processed, inevitably leading to misinterpretation, a process analogous to the loss of energy when impedances are not matched.
electricity comes in two major varieties: direct current (DC), beloved of Edison, in which electricity flows in a continuous fashion like a river, and alternating current (AC), in which it flows in a pulsatile wave motion much like ocean waves or the blood in your arteries.
one can take advantage of its pulsatile nature and match impedances at branch nodes in the power grid so as to minimize power loss, just as we do in our circulatory system.
the nature of the flow makes a transition from being pulsatile in the larger vessels to being steady in the smaller ones. That’s why you feel a pulse only in your main arteries—there’s almost no vestige of it in your smaller vessels.
this leisurely speed ensures that oxygen carried by the blood has sufficient time to diffuse efficiently across the walls of the capillaries and be rapidly delivered to feed cells.
the theory predicts that these velocities at the two extremities of the network, the capillaries and the aorta, are the same for all mammals,
But what’s really surprising is that blood pressures are also predicted to be the same across all mammals, regardless of their size.
Most biological networks like the circulatory system exhibit the intriguing geometric property of being a fractal.
fractals are objects that look approximately the same at all scales or at any level of magnification.
This repetitive phenomenon is called self-similarity and is a generic characteristic of fractals.
the fractal nature of the circulatory system subtly changes from the aorta to the capillaries, reflecting the change in the nature of the flow from pulsatile to nonpulsatile.
The space-filling requirement that the network must service the entire volume of the organism at all scales also requires it to be self-similar in terms of the lengths of the vessels.
