This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Some critical trends in information theory, its role in living systems and utilization in fluctuation theory are discussed. The mutual information of thermodynamic coupling is incorporated into the generalized fluctuation theorem by using information theory and nonequilibrium thermodynamics. Thermodynamically coupled dissipative structures in living systems are capable of degrading more energy, and processing complex information through developmental and environmental constraints. The generalized fluctuation theorem can quantify the hysteresis observed in the amount of the irreversible work in nonequilibrium regimes in the presence of information and thermodynamic coupling.

Definition and quantification of information have created broad discussions; particularly, “information theory” in living systems is an evolving field [

In conventional thermodynamics, the amount of entropy is independent of how the process is regarded as being divided into irreducible subsystems; the entropy of a system can be calculated from the entropies of its subsystems. If the subsystems are statistically independent that is not correlated by the mutual information, the entropy is additive. Shannon’s entropy of the total system is given by the sum of those of the subsystems [

An over damped motion _{0}. For an arbitrary number of degrees of freedom,

For a system in contact with a heat bath, symmetry of the probability distribution of entropy production in the steady state is known as the fluctuation theorem. Crook’s fluctuation theorem compares probability distributions for the work required in the original process with that of the time-reversed process. The probabilistic approach has gained broader appeal due to the advances in experimental techniques, such as atomic force microscopes and optical tweezers, for tracking and manipulation of single particles and molecules [

Information may be defined as the capacity to reduce statistical uncertainty in the communication of messages between a sender and a receiver [_{i}

Probability _{i}_{i}

where the occupation frequency of state _{i}_{i}

In Shannon’s theory, entropy represents the amount of uncertainty one particular observer has about the state of the system [_{1}, _{2},…, _{N}_{i}_{i}_{i}_{5} then the uncertainty about

If we define another random variable _{1}, _{2},…,_{M}_{1}, _{2},…, _{M}

Here “:” shows that information is symmetric;

For independent variables

The equation above shows that information measures deviation from independence that is the amount by which the entropy of

Maximum entropy production (MEP) may be an organizational principle applicable to physical and biological systems. Nonequilibrium thermodynamics and MEP can be used to derive overall evolutionary trends of Earth’s past; for example, they can identify the role that life plays in driving thermodynamic states far from equilibrium [

Implementation of the MEP principle also includes the steepest entropy ascent formalism based on quantum thermodynamics approach [

Consider _{i}(t_{ij}_{ji}_{ij} = c_{ji},_{ij}_{ij}_{ij}_{i,a}_{i,a}_{j} J_{ij,a}_{ij,a}_{t} J_{ij}_{i}_{a}_{ij}_{a}p_{a}J̄_{ij,a}

with the constraints [

the most likely probability on the path space is estimated as:

where _{ij}_{ij}_{a}_{a}_{i} λ_{i} n_{ia}_{ij} n_{ij} J̄_{ij,a}_{i}_{ij}_{ji}

However, a trajectory of a quantity possessed by a system may fluctuate wildly (if far from equilibrium) or weakly; than they would not have the same probabilities as long as they have the same initial and final states. Here the path trajectory

where

The partition sum and the constitutive equation of motion have the relations:

The forward and backward components of the time and ensemble averaged fluxes are:

where _{ij}c_{ij}

The entropy production of a microscopic path

By using the most likely probability on the path space _{a}

where

The entropy curvature (response) matrix and the probability distribution for the time averaged flux

In near equilibrium regime that is for small

The first part on the right side of the equation above is the logarithm of the total number of paths for uniform probability distribution, while the second term is the entropy production. In the MEP, the assumption is that the number of paths _{I}

MEP principle states that if thermodynamic force _{i}_{i}_{τ}

yields the maximum value of the σ_{τ}

In the meantime, the relationship between the fluxes and forces:

indicates that this relationship can be both linear and nonlinear [

The same entropy production can be both maximum and minimum depending on the constraints used in the entropy production variation. However, it is widely published that the MEP principle may be a critical link in the explanation of the direction of the biological evolution under the imposed constraints of the environment [

In the cortex, populations of neurons continuously receive input signals from other neurons, interpret their ongoing activity, and generate output signals destined for other neurons. This information processing and transmission is limited by the repertoire of different activated configurations available to the population. The extent of this repertoire may be estimated by its entropy _{x}_{n}_{x}_{n}_{x}_{n}

For a number of unique binary patterns, the entropy is:

where _{i}_{C}

where Ω is the number of possibilities,

Genes are sequences of deoxyribonucleic acids (DNA) nucleotides composed of four bases (guanine G, adenine A, cytosine C, thymine T). Genes carry and transmit the information to specify 20 amino acids for protein synthesis. The genome refers collectively to the total genetic information coded in a cell. Ribonucleic acid (RNA) molecules transfer information from DNA to the site of protein synthesis [

During transcription, one of the strands of the DNA acts as a template and determines the sequence of RNA nucleotides. RNA also contains the base uracil U in place of thymine T. The transcribed RNA binds to ribosomes in the cytoplasm where proteins are synthesized from the encoded information. Only exon segments actually code for amino acids, and a continuous sequence of exons becomes mRNA, which binds to a ribosome. In translation, each ribosome is composed of proteins and ribosomal RNA called rRNA. Transfer RNA (tRNA) is the link between an amino acid and its mRNA codon. Each amino acid brought to a tRNA molecule is linked by a peptide bond to the end of the growing protein chain. The completed protein then undergoes folding. Signals from within or outside the cell can turn on or off the transcription of genes. Homeostatic control systems regulate compensatory responses in which the set point of a variable is controlled. In case of a continuing perturbation, the error signal (negative or positive feedback) may lead to resetting the set point [

A stimulus is a detectable change in the internal or external environment. A receptor detects the stimulus and produces a signal that is relayed through various pathways to the integrating center, which receives signals from many receptors. The output of the integrating center reflects the net effect of the total afferent input. The output is sent through the efferent pathway to an effector.

Some of the intercellular chemical messengers are hormones, neurotransmitters, and paracrine agents, which are synthesized by cells and diffuse to target cells upon receipt of an appropriate stimulus. Autocrine agents are local chemical messengers that act upon the same cells that secreted them. Many homeostatic systems add and remove different chemicals in response to a particular stress; it is mostly reversible without the involvement of genetic change.

Receptors are glycoproteins on plasma membranes of target cells and bind specific messengers. The binding leads to response through the signal transduction pathways that would vary in different cell types. For example, neurons generate electric signals that pass from one cell to another and release chemical messengers called neurotransmitters to communicate with other cells. A synapse is a junction between two neurons and its influence can be either excitatory or inhibitory. If many presynaptic cells affect a single postsynaptic cell it allows information from many sources to influence the activity of one cell. On the other hand, if a single presynaptic cell affects many postsynaptic cells it allows one information source to affect multiple pathways [

DNA is a code, and codes from sequence alone do not reveal information. DNA replication starts with an initiator protein, which unwinds a short stretch where another protein known as “helicase” attaches to and breaks apart the hydrogen bonds between the bases on the DNA strands. The nonconditional entropy for DNA sequence or proteins is about two bits per base; a random protein would have log_{2}(20) = 4.32 bits of entropy per site. Due to repetitions, pair, and triplet correlations the actual entropy would be lower [

In equilibrium thermodynamics, isolated systems have the maximum entropy and there are no correlations between the states; hence there is no information. The information as the amount of correlations between two systems stored in living system (biological genomes) points out that they are far away from equilibrium. Consequently, the information theory becomes a part of nonequilibrium thermodynamics in living cells. Information measures the amount of entropy shared between two systems. Also, information enables us to make predictions about other systems, since only in reference to another ensemble entropy can become information. Therefore, what is described by the correlations between the sequences stores information not the sequence itself. On the other hand, what information a genomic sequence represents depends on the interpreter environment. Therefore, if a sequence means something it can create a function (response) necessary for its environment [

The information theory introduced “functional information” that leads to self-organizing capabilities of living systems, and “instructional information” that is a physical array. However, linkages with the field of semiotics established a much more compatible approach to biological information [

Functional information _{x}_{x}_{x}_{2}[_{x})], where _{x}_{x}

Instead of a single RNA nucleotide, a minimum sequence length (_{min} nucleotides) would achieve any significant degree of ribozyme function, _{x} > E_{min}. Increasing the number of nucleotides (_{min}) will generally lead to more functional sequences. Consequently, for a maximum possible degree of specific function _{max} there will be an optimal RNA sequence of length _{opt} possessing the maximum functional information of _{max} (_{max}) = −log_{2} [1/(∑_{1−}_{nopt}^{n}_{x}_{max}, an intermediate functional information shows [_{x}_{max}(_{max})] [

Each position on the genome is four-base code and the uncertainty at each position is two bits; then the maximum entropy becomes:

since _{j}_{j}_{j}_{j}

where

The thermodynamics of protein structures implies that sequence and structure are related. If a structural entropy of proteins

where

Therefore, thermodynamic entropy of a protein structure is limited by the amount of information about the environment coded by the sequence. This may imply that sequences that encode more information about the environment may be more functional.

One of the consequences of the Human Genome project has proved that biology is an “informational science” [

In semiotic understanding of living systems, interpreters of signs and information will often be an interpreter-dependent objective process. Genes should be regarded as signs in DNA, which can only have any effect on a cell function through a triadic-dependent process that is the communication of a form from the object to the interpretant through the mediation of the sign [

There are two aspects of molecular and supramolecular information transduction: (i) sphingomyelinase activity leading to a lipid-mediated cross-communication between the sphingomyelinase and phospholipase A_{2} pathways, and (ii) the compositionally driven lateral organization of whole glial and neuronal membrane interfaces leading to the differential responses. The interfacial properties may express and regulate the generation, transduction, and amplification of molecular and submolecular properties on a temporal and structural scale mainly at mesoscopic level. However, the transduction of information mainly occurs between the microscopic (<10^{−6} s, <1 nm) and the macroscopic (>10^{−1} s, >500 nm) ranges [

A physiological mechanism may exist to ensure fast transmission of information to that hemisphere which is more efficient in its processing [

Signal transduction includes the interrelation and information exchange between phosphohydrolytic pathways producing lipid mediators of signal transduction whose formation modifies the interfacial composition. At the molecular level, there exists cross-talk between a glial cell and the axonal membrane surface. There also exists the subtle intercommunication and response of functional living cells to information contained in the molecular organization of contacting cellular membranes [

Thermodynamic coupling [_{i}J_{i}X_{i}_{1} < 0 (unnatural) is thermodynamically coupled with another process with σ_{2} >> 0 (natural) to achieve a desired output function; therefore, the coupled system satisfies the second law of thermodynamics [(σ_{1} < 0) + (σ_{2} >> 0)] > 0. Here, the thermodynamic coupling refers that a flux

Biochemical reactions coupled with diffusion of species can lead to molecular pumps and biochemical cycles, such as the adenosine triphosphate (ATP) synthesis coupled to the respiratory electron transport. This shows a functional process leading to organized structures where the ATP synthesis (σ < 0) has been made possible and the coupled processes satisfy ∑

The general approach for incorporating thermodynamics into the information theory has been to derive probability distributions for nonequilibrium steady states by employing the variational principle. However, composing the appropriate constraints to be used in the variational principle is not clear, since there is no definite extremum quantity to characterize the state space of such steady nonequilibrium states. In the vicinity of equilibrium only, the linear phenomenological laws may be useful in that respect [

The unified theory of evolution attempts to explain the origin of biological order as a manifestation of the flows of energy, matter, and information on various spatial and temporal scales. Genes originates the information to synthesize the required enzymes, regulatory and structural proteins. The genome is the source of cellular codes; also any cellular structure such as lipids and polysaccharides may store and transmit information. Besides, thermodynamic forces in the form of transmembrane gradients of H^{+}, Na^{+}, K^{+}, Ca^{2+} and consequent electric potential cause significant displacements from equilibrium, and are therefore, potential sources of information. Genome-protein system may be a component of a large ensemble of cellular structures, which store, encode, and transmit the information [

Le Chatelier’s principle may be applied to analyze how a protein-signaling network at equilibrium returns to its equilibrium state after being slightly perturbed [

The composite immediate object of a protein coding gene is the sequence of amino acids of a polypeptide, which can be folded in different ways in different cellular contexts and represents dynamical objects. So sign is a sequence of nucleotides in DNA and becomes effective information only when it is used by an active, coordinated, and coupled system of interpretation in the cell. This means that a range of sequence of amino acids required by the environment (cell) has been constructed [

The use of maximum entropy formalism in biology is expanding [

For a multicomponent fluid system under mechanical equilibrium with _{r}_{τ}

where _{i}_{i}_{i}μ_{i}

For a steady state system, we have ∇ · _{i}_{i,j}ν_{ij}J_{rj}

Considering _{i}_{ik}X_{k}

Onsager’s reciprocal relations state that the coefficient matrix ^{2}−^{2}−

Thermodynamic coupling may lead to (^{2}−

Fluctuation theorem (FT) relates the probability _{τ})_{τ}_{τ}

where the Boltzmann constant _{B} = 1. The probability distribution of the rate of entropy production becomes _{τ}_{τ}_{τ}

Crook’s FT can be used to determine free energies of folding and unfolding processes occurring in nonequilibrium systems. For that, the unfolding and refolding processes need to be related by time-reversal symmetry,

In processes that are microscopically reversible, Crook’s FT predicts a symmetry relation in the work fluctuations for forward and reverse changes as the system is driven away from thermal equilibrium by the action of an external field or perturbation. A consequence of Crook’s FT is Jarzynski’s equality exp(−Δ_{B}_{B}

Sagawa and Ueda [

In the absence of the initial of final correlations, entropy production satisfies the integral of FT (or Jarzynski’s equality): 〈(−

Consider a typical information processing in a composite and stochastic system X, which is in contact with multiple heat baths (_{k}_{B}) and interacting with system Y, which does not evolve in time during the interactions. The time evolution of X depending on the state of Y from _{i}_{f}_{ia}_{i}_{fa}_{f}_{i}_{a}_{a}_{ia}_{i},y_{fa}_{f},y_{ia}_{i}_{ia}_{i,}y_{fa}_{f}_{fa}_{f},y_{a}_{ia}_{i,}y_{fa}_{f},y_{ixiy}_{fx,fy}

In the presence of information

where Δ_{fxfy}_{ixiy}_{ia}_{i},y_{rem}) by system

where 〈_{rem}〉 may be an upper bound of the correlation that can be used.

The detailed FT in the presence of information processing becomes [

with the constraint _{b}_{f}_{i}_{f}

FT allows a general orthogonality property of maximum information entropy MIE to be extended to entropy production EP. Maximum entropy production MEP and the FT are generic properties of MIE probability distributions. Physically, MEP applies to those macroscopic fluxes that are free to vary under the imposed constraints, and corresponds to the selection of the most probable macroscopic flux (flow) configuration [_{a}_{τ}_{n}_{n}_{-n}_{n}_{-n}_{B}_{B}

_{ik}, i

where _{TC} represent the energy cost of information introduced because of thermodynamic coupling. In _{TC}

Various aspects of the thermodynamic relationships between information theory and living systems are discussed with emphasis on some new efforts toward defining and estimating the information. These efforts may lead to a deep understanding of the role played by the information particularly in living systems. Information theory can be used in investigating protein-protein interactions and the association of enzymes and proteins with their binding sites. If the subsystems are statistically dependent that is correlated by the mutual information, then the entropy becomes nonadditive; however, entropy production of the subsystems are additive. By treating mutual information and entropy production on equal footing a new integrated fluctuation theorem has already been suggested by Sagawa and Ueda [

The author declares no conflict of interest.