Biology Reference
In-Depth Information
If the number of input trains carrying the leaves is
m
, the number of coincident input
trains is
j
, and the probability of the input signal being in one time bin is
p
, we can
use the Mikula-Niebur equation (
7.18
), to calculate the probability P of observing
output 1 (Fig.
7.8
) at a time point as follows:
m
j
m
C
j
p
j
P
¼
ð
1
p
Þ
(7.19)
where
m
C
j
is the binary coefficient given by m!/j!(m
j)!.
Each node in the temporal hierarchy represents an event; the higher its position
in the hierarchy, the less probable or rarer is its occurrence. The probability of
output P in Eq.
7.19
is a nonlinear function of three variables,
m
,
j
, and
p
, assuming
that the correlation coefficients among the input signal trains are zero (Mikula
and Niebur 2003). Therefore, there may be an optimal set of variables for
maximizing the P value. If the Mikula-Niebur equation (
7.18
), can be applied to
multiprotein complexes with r subunits divided into two groups, one group
consisting of s coincidence detectors with very short
t and the other of l (lower
case L) coincidence detectors with very long
D
t, so that r
¼
s + l, then s detectors
will affect P through p, and l detectors will affect P through C in Eq.
7.19
.
Equation 7.19 may provide one possible rationale for the existence of multiprotein
complexes. In other words, the multiplicity of the components in a multiprotein
complex can enhance the probability of the rare event (i.e., node 1 in Fig.
7.6
)
outputted by a system of coincidence detectors (represented by all the nodes having
3
of connectivity) by either increasing C or optimizing p in Eq.
7.19
.
If a temporal hierarchy consists of n levels (an example of a five-level temporal
hierarchy is given in Fig.
7.8
), there will be 2
(L
1)
nodes at the Lth level, for
example, 2
4
D
5, that is, Nodes 16 through 31. For convenience,
we will define the order of a node as (n
¼
16 nodes at L
¼
L), where n is the total number of the
levels constituting the temporal hierarchy under consideration and L is the level of
the node. We will refer to the nodes at the Lth level as the (n
L)th-order
coincidence detecting events (CDEs). Thus, Nodes 16 through 31 represent the
zeroth-order CDE, 8 through 15 represent the first-order CDEs, 4 through 7
represent the second-order CDEs, etc. The higher the order of a CDE, the smaller
would be the probability of its occurrence in nature since its occurrence depends on
the occurrences of its lower order CDEs, unless facilitated by some mechanisms
driven by dissipation of free energy. In other words, we can recognize two kinds of
CDEs -
passive
and
active
CDEs. The temporal hierarchy (TH) built from
passive
CDEs may be called
passive TH
and that build up of active CDEs as
active TH
.
I here postulate that living systems are examples of
active THs
and
passive THs
belong to abiotic systems.
The idea that living systems embody active THs seems to be supported by both
theoretical and experimental evidences:
1. Enzymes are active coincidence detectors in that the simultaneous localization
of three or more catalytic residues at the active site of an enzyme at the transition
state, driven by the free energy of substrate binding, leads to catalysis as
indicated in Fig.
7.4
.
