Perhaps, you are thinking of Einstein’s
theory of relativity when you are reading the title of this
article. However, what you are about to learn is nothing to
do with the space-time traveling as in the episodes of Star
Trek but related to the common phenomena of life. So, what
exactly is the 4th dimensional of life meant? Before we give
an answer to this question, let’s take a look at how
we can measure the dimension of an object.
It is known that some physical properties of
an object will vary with the physical size of the object.
For example, the weight of sphere will increase with its radius.
However, have you ever thought about that these variations
are related to the spatial dimension of the object? A common
example to illustrate this point is the relation between the
radius (R) of a sphere and its mass (M). If the sphere is
made of a material with density ?, from our text book, we
know that:
or 
or 
(1)
This formula says that: the mass of the sphere
is proportional to the third power of its radius. If we do
not know how to derive the above relation, we can easily check
the third power dependence with experiments. The importance
of the above relation lies in the numerical value 3 of the
 .
It says that the sphere is a 3 dimensional object. Therefore,
our world must be at least 3 dimensional. For a man living
in 2 dimensions (2D), a “sphere” is a circular
disk. If this 2D man has no formula in his text book to tell
him about the relation between the mass of the disk and the
radius of the disk, he can also determine by experiments that
the mass of the disk is proportional the second power of the
radius of the disk ( ).
Therefore, a 2D main can deduce that the disk is a 2D object
by the numerical value 2 in .
Relations similar to the above formula are common among different
physical quantities of an object or system. They are called
scaling relations. As you might have suspected, the most important
quantity in a scaling relation is the exponent which is called
the scaling exponent. It tells us how properties of the system
change with the size of the systems. Of course, it is related
to the physical dimension of the system as we have seen in
the above example. Since we are living in a three dimensional
world, exponents similar to those mentioned above with values
such as (+-)3n or (+-)m/3 often appear in physics with n and
m being integers.
After we have learned how to measure the physical
dimension of a physical system by using scaling relation,
we might want to know if such methods can be applied to other
non-physical systems such as living systems. Also, if the
scaling method works, what does the dimension mean? In the
above example, we have chosen the weight or mass of the sphere
as the scaling properties because we know that they are closely
related to the size of the system. We cannot choose other
quantities such as temperature, color or density of the sphere
as scaling properties because these quantities are not related
to the size of the system. For biological systems, what are
the important system characteristics which will vary with
the system size? There are no unique answers to the last question
because there are many characteristics of a biological system
which can vary with the system sizes. For example, the amount
of food in-take, the speed of movement and even the life-span
of animals can all be size dependent. However, in general,
all living systems need metabolism. All of these quantities
are closely related to the metabolic rate of the systems.
Thus, metabolic rate is perhaps a good quantity to be used
in a scaling method to measure the dimension of a living system.
Now, let us learn how to measure the dimension
of biological systems from the rate of heart beats (an indicator
of metabolism) of animals. For example: the rate of heart
beat of a mouse is about 400~500 beats per minute and those
of chicken, dog, human and elephant are 200, 100, 70 and 30
beats per minute respectively. It seems that the larger the
animals are the slower are their rates of heart beat. This
relation can be illustrated in graphical form as shown in
Figure 1 where the beat rate (f) is plotted as a function
of the body weight of the animals. From the log-log form of
Figure 1, we have:
Rate of heart beat : f ~  (2)
The straight line in Figure 1 is the plot of
relation (2) with b = -0.25. One has to be careful about the
meaning of the data shown in Figure 1 as the nature of the
data are differently from those commonly used in physics.
For example, the data in Figure 1 are collected from the Internet
/1/ where the rate of heart beat and weight of a human are
taken as 70 times per minute and 70 Kg respectively. However,
it is known that both the heart beats and body weight are
not fixed. The 70 times per minute and 70 Kg are just the
statistical averages taken from a sample of healthy people.
Of course, different statistics will give different results.
However, it is found that these data can always be expressed
by a relation similar to (2) but with different values of
b; such as -0.27 or -0.23. In general the values of b from
different statistics are similar to the result of Figure 1;
not far from -0.25.
(Figure 1)
Data in Figure 1 come from different animals.
Since they are not directly related, it is quite amazing to
find a relation such as (2) even it is statistical in nature.
Either there are similar mechanisms in the controlling of
heart beat among these animals or a more fundamental reason
is behind the observation in Figure 1. Similar relations have
also been observed and reported in plants. For example, it
is known that the distribution density of plants is related
to the mass of the plants. Other common examples are the rate
of food intake in animals and the rate of fluid transport
in plants. These quantities are all related the mass of the
systems in form of a power-law similar to (2). If life phenomena
can be directly explained in terms of physical laws, b must
be related to (+-)3n or (+-)m/3 as one might expect. However,
a large number of measurements has suggested that b is closer
to (+-)4n or (+-)m/4. That is: b is related to 4 not to 3
but there are also some exceptions. The relation between the
important characteristics of the system and the system size
is usually referred to as allometric scaling.
Since statistical error in biological experiments
are far larger that those found in physics experiments, you
might wonder how do we know for sure that the exponent b is
related to 4? Currently, we do not have a fundamental answer
to this question. However, a model based on the transport
of fluid in a plant proposed by Prof. West has demonstrated
that the number 4 comes from a relation: b = (D+1) = 4 where
D = 3 is the dimension of our physical world. The basic assumption
of this model is that living systems are optimizing their
metabolic rate of the systems by efficient transports of fluid
(nutrition) to all parts of the systems. To achieve this goal,
a transport system with unique geometry is developed by biological
systems. For example, both our blood circulation system and
the fluid transport system in plants are made of branching
tubes interconnected from large to small scales; distributed
throughout the body as shown in Figure 2.
(Figure 2)
The geometric shapes shown in Figure 2 are
called fractal which stresses that the dimension of the system
is not an integer but a fractional number. An important characteristic
of the fractal structure shown in Figure 2 is that it is self-similar.
If we magnify a small part of the structure shown in Figure
2, we will find similar structure as the original structure
in the magnified part. That is to say: the same structure
is repeated on different length scales. In Chinese folklores,
there is also a self-similar story as:” A long time
ago, there was a mountain far away. An old monk and his disciple
were living in a temple in the mountain. The old monk was
telling a story to the young monk as:” A long time ago,
there was a mountain far away. An old monk and his disciple
were living in a temple of the mountain. The old monk was
telling a story to the young monk as:” A long time ago,
…””. This story is self-repeating.
Since Prof. West’s model can be used to
explain successfully the value of 4 and some other related
problems, he believes that life endows a new and unique dimension
to living systems which change from D = 3 of non-living systems
to D = 4 for living systems and called it the fourth dimension
of life. Because of that, he also calls for experimental investigation
of quasi-2D biological systems to see if D = 3 can be found.
If the D = 4 of living systems originates from
the optimization of design, is this unique to biological systems
only? Can we observe D = 4 in other non-living systems? Prof.
West gives a negative answer. He takes the simple example
of an engine of a car to explain his idea. Obviously D = 4
cannot be observed in an engine which means: if we plot the
“metabolic” rate of the engine such as rotations
per minute, fuel consumption or output power as a function
of the weight of the engine, we will not get a result similar
to (2). This conclusion is reasonable because even the most
optimized engine must still obey the law of physics. Its "physiological"
parameters are only design parameters of the engineers. There
are no reasons why they should behave as predicted by (2).
Obviously, when compared to biological systems, an engine
is far too simple. Also, different parts of a biological system
are believed to evolve to their present forms not by design.
(Figure 3)
A more realistic example of artificial structures
is an airplane. It is probably the most complex artificial
system created by man. Although an airplane is also designed
by engineers using physical laws, the complexity of the structure
of an airplane is far higher than that of an engine and could
not have been designed by only a few engineers. Currently,
an airplane is consisted of many complex but independent systems.
A Boeing 777 requires more than 1000 computers and 150,000
modules interconnected to coordinate its flight. Obviously,
the design of such a plane is not started from scratch from
a few engineers. Similar to evolution in biology, the design
and building of such a plane is through numerous improvements
over a long time; starting from a very simple plane. During
this evolution process, different parts of the plane will
"interact" to improve the overall performance of
the plane. In this aspect, we can regard the plane as a more
or less living entity. We can also compare the performance
of an airplane to those of flying animals. Figure 3 is the
result of such a comparison. It can be seen that the cruising
speed of a flying system including the Boeing 777 is related
its weight similar to (2). With this last result, one can
really think of an airplane as a living machine! Data in Figure
3 come from a very interesting book /3/ which discusses many
aspects of flying in simple terms.
From the above example, we can see that even
for non-living system, if it is complex enough so that different
parts are evolving with interactions to improve the performance
of the system, it is possible that the system will show behaviors
similar to those find in (2). The picture emerges is that
an originally relatively simple physical system can evolve
into a structure with complexity and characteristics similar
to those of living system by optimization interactions. We
will then tempt to ask: will we understand the origin of life
by studying these artificial complex systems? Also, since
all these systems are built on physical laws, can the phenomena
of life be understood just by physics alone? Of course, there
are still no clear answers to these very fundamental questions.
Finally, it must be pointed out that these "improve
and evolve" process in artificial systems are not the
physical laws that we know of. They are not intrinsic to the
systems but are added artificially by human in the quest for
better performance. Currently, there is a new discipline in
physics called the artificial life (AL). Many people working
in the area of AL believe that laws of physics alone will
explain all the phenomena of life. They are working hard to
find a spontaneous mechanism for the "improve and evolve"
process. If this goal can be achieved, we might be not far
from the answer to the question of the origin of life.
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