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Appendix 4: General Theory of Molecular Machines

A receiver is a device whose state is determined by an external signal. In contrast, a simple molecular machine such as EcoRI is not directed to its after state (binding sites) by an external command. Encoding or decoding a communications signal also requires a memory to record the signal as it is being processed. Simple molecular machines don't have the necessary memory. For example, DNA in the groove of EcoRI acts like a key in a lock, with the recognition process taking place in parallel over a surface of contact between EcoRI and DNA [Rastetter, 1983,Gilbert & Greenberg, 1984,McClarin et al., 1986]. Since EcoRI has no record of its previous bound and unbound states it has no record of its history and cannot handle a time varying communications signal.

However, a time-encoded message could be received, remembered and processed by a combination of simple molecular machines. Such a ``molecular receiver'' could decode a message of the kind that Shannon's theory is designed to handle. Since they could be made insensitive to thermal noise by appropriate coding, molecular receivers are likely to play an important role as the interface between humans and artificial molecular machines and molecular computers. It is not known if any living organisms contain such devices, although the processes of translation, cell movement, mitosis, embryonic development and circadian rhythms are candidates.

According to Fourier analysis, a time varying signal may be recorded as a series of discrete samples. If t is the period of the recording and W is the highest frequency in the signal's spectrum, then the original signal may be reproduced exactly if at least

 

d_time= 2 t W (49)

samples are recorded [Shannon, 1949,Conway & Sloane, 1988,Walker, 1988]. This powerful result is the basis of digital-sound recording methods such as the compact disk [Walker, 1988].

If distinct states of a molecular receiver are determined by an external communications signal, then a high dimensional space consisting of

 

D = d_spaced_time (50)

dimensions can be used to describe the coding space of the machine. The machine could take advantage of both the spatial and the time dimensions and would operate in a ``space-time'' we will call Z space.

As in equation (31), we find that the average total energy for the entire molecular receiver in Z space is

 

$$\langle E_z \rangle ({\scriptstyle \frac{1}{2}}k_{\mbox{\scriptsize B}}T) \times D$$ =

$$t d_{space}(Wk_{\mbox{\scriptsize B}}T) \;\;\;\;\;\;\mbox{(joules)} .$$ = (51)

using equations (49) and (50). Dividing both sides of (51) by t gives the total thermal noise for the molecular receiver:

 

$$\equiv \frac{\langle E_z \rangle}{t}$$ N_z

$$d_{space}(Wk_{\mbox{\scriptsize B}}T) \;\;\;\;\;\;\mbox{(joules per second)} .$$ = (52)

The probability density is still given by equation (48). The sphere volume, which gives the capacity, depends on the radius raised to the dimension that the sphere is embedded in, so the maximum number of states is:

$$M_z \leq \frac{ V_{before} } { V_{after} } = \left( \sqrt{ \frac{P_z + N_z}{N_z} } \; \right) ^{d_{space}2 t W} .$$ (53)

The definition of the molecular receiver capacity follows Shannon's definition exactly [Shannon, 1949]:

$$C_z = \frac{\log_2( M_z)}{t} = d_{space}W\log_2 \left( \frac{P_z+N_z}{N_z} \right) \;\;\;\;\;\mbox{(bits per sec).}$$ (54)

The relationship of this general equation to the capacity equations in the other two theories is straightforward. If we set d_space= 1 to indicate that there is only one spatial degree of freedom, we obtain Shannon's formula (equation (45)), and equation (52) becomes Nyquist's formula for thermal noise in a single wire [Nyquist, 1928,Johnson, 1928,Pierce, 1980]. If instead we set t W= 1(to indicate a complete lack of long-term memory) and use the time independent capacity definition $C_z = \log_2(M_z)$, we obtain the formula for a simple molecular machine, equation (38), and the thermal energy formula (30) is obtained from (51).

The three theories are summarized in Table 1.

 

Table 1: Information Capacity Theories

	Channel	Molecular Machine	Molecular Receiver
Coding Space	Shannon	Y	Z
Degrees of Freedom	$d_{time}= 2 t W$	$2 d_{space}\le 2 (3n - 6)$	$D= d_{space}d_{time}$
Power	$P$	$P_y$	$P_z$
Noise	$N = Wk_{\mbox{\tiny B}}T$	$N_y = d_{space}k_{\mbox{\tiny B}}T$	$N_z = d_{space}Wk_{\mbox{\tiny B}}T$
Power & Noise units	J / sec	J / op	J / sec
Capacity	$C = W\log_2(P/N + 1)$	$C_y = d_{space}\log_2(P_y/N_y + 1)$	$C_z = d_{space}W\log_2(P_z/N_z + 1)$
Rate	WsR	R	WsR
Capacity & Rate
units	bits / sec	bits / op	bits / op - sec

 The units are J: joules; sec: seconds, op: operation.

The capacity of a molecular receiver is most easily understood as the capacity of d_space parallel communications channels (compare (45) to (54)). The method of encoding in space would then correspond to spreading the coding bits across the parallel channels rather than spreading them out over time. From this it is clear that for a given error rate one can reduce the required encoding and decoding time by increasing the parallelism.