# Entropy Generation and Control: Criteria to Calculate Flow Optimization in Biological Systems

^{1}

^{2}

^{*}

## Abstract

**:**

^{Jσ}). We demonstrate the restricted possibility to obtain an increase in flow along with a decrease in entropy generation, and the more general situation of increases in flow along with increases in entropy generation of the process. In this scenario, the C

^{Jσ}aims to identify the best way to combine the gain in flow and the associated loss of useful work. As an example, we analyze the impact of vaccination effort in the spreading of a contagious disease in a population, showing that the higher the vaccination effort the higher the control over the spreading and the lower the loss of useful work by the society.

## 1. Introduction

_{co}is the cardiac output (the flow), X

_{P}is the pressure difference between the arterial and the venous trees (the unbalanced force) and G

_{per}is the peripheral vascular conductance (the inverse of the peripheral resistance). From the control engineering standpoint, the cardiac output is a regulated variable while the pressure difference and the conductance are controlled variables [4,5]. In this sense, pressure and conductance are directly under a certain policy rule to regulate the flow, which is not directly accessed by the controlling system.

## 2. Flows and Optimization

## 3. Thermodynamic Criteria for Optimization

## 4. The Present Approach to Optimization

_{i}(A) of a process “i” under condition “A” is lead to J

_{i}(B) under condition “B”, does this follows an optimal path? Note that we are not asking whether J

_{i}(A) or J

_{i}(B), or both, are extrema. We are concerned about how optimal the transition between conditions is.

^{ibbs}, is given as:

_{final}< C

_{initial}. This $\u2206{\mathrm{G}}^{\mathrm{i}\mathrm{b}\mathrm{b}\mathrm{s}}$ equates, then, with the entropy increase as $\mathrm{T}\u2206\mathrm{S}=-\u2206{\mathrm{G}}^{\mathrm{i}\mathrm{b}\mathrm{b}\mathrm{s}}$. On the other hand, if the initial concentration in the system is kept constant despite the outflow of molecules, then the time derivative of the lost work is the most useful power in the system:

_{2}and C

_{1}are fixed values. Note that $\frac{\mathrm{d}\mathrm{n}}{\mathrm{d}\mathrm{t}}$ is the flow of molecules and the rest of the equation to its right side is the chemical potential. From this example, we now proceed to generalizations.

#### 4.1. Entropy Generation

#### 4.2. Coefficients of Control

**K**= {k

_{1}, k

_{2}… k

_{n}} in such a way that, for a given process “i”:

_{a}∈

**K**. Then, flow regulation by the parameter k

_{a}implies in:

_{a}) [34,35,36,37,38]:

_{a}:

_{a}of the system and that it is in respect to a certain process (or flow). Taking for granted that these are implied in the notation, we simplify the flow-entropy coefficient to:

_{i}in the total entropy generation of the system as a non-reversible heat loss to the surroundings. In a vast number of cases, within the time scale of the changes in a given flow, the surroundings can be taken as an infinite pool and, thus, one can consider ${\mathrm{C}}_{{\mathrm{k}}_{\mathrm{a}}}^{\mathrm{T}}=0$ for the purposes of the analysis. This ${\mathrm{C}}_{{\mathrm{k}}_{\mathrm{a}}}^{\mathrm{T}}=0$ will be assumed in the remaining of the present text, except in the working example at the end. Then, Equation (13) becomes:

_{a}.

#### 4.3. The Values of ${C}^{J\sigma}$

_{a}, the associated ${\mathrm{C}}^{\mathrm{J}\mathsf{\sigma}}$ value is a single point in the ${\mathrm{C}}^{\mathrm{G}}\mathrm{v}\mathrm{s}.{\mathrm{C}}^{\mathrm{X}}$ plane. Alternatively, if ${\mathrm{C}}^{\mathrm{J}\mathsf{\sigma}}$ turns out as a function of the parameter, then it might assume different values along the changes in k

_{a}. In the final section of the present manuscript, we develop a case study of this last type.

#### 4.4. ${C}^{J\sigma}$ and Physiological Adjustments

_{a}, and that the associated coefficient is ${\mathrm{C}}^{\mathrm{J}\mathsf{\sigma}}=0.8$. From Figure 3, we can see that this is a fairly good value representing an increase of 0.8 units of entropy production for each unit of increase in the flow; that is, the proportional increase in entropy generation falls below the increase in flow. Then, the system returns to condition A, decreasing the parameter k

_{a}. Since ${\mathrm{C}}^{\mathrm{J}\mathsf{\sigma}}$ is the same, the returning path also has a decrease in entropy generation that falls behind the decrease in flow. That is, while in the first transition, the system obtained an advantage in terms of the relationship between the increase in flow and the increase in entropy generation, in the returning transition the system has a disadvantage in the decrease in flow and the decrease in entropy generation.

_{a}, this is not possible. In terms of phylogenetic evolution and ontogenetic development this is not a problem since these are one-way paths that once traveled would be no return. However, for physiological adjustments, the picture is quite different because they are two-way roads.

_{a}attains specified values, the intermediate conditions B’ and A’ comprise changes through other parameters that allow for hysteresis in the cycle. Then, as in the Carnot cycle, the area enclosed by the path represents a net gain in terms of entropy generation in relation to changes in flow.

## 5. A Working Example

^{−1}] and the coefficients are non-dimensional quantities, indeed.

^{G}vs. C

^{X}plane (Figure 3): if v = 0, ${\mathrm{C}}_{\mathrm{v}}^{\mathrm{J}\mathsf{\sigma}}=2+\mathrm{z}\xb7\frac{\mathrm{a}\xb7\mathrm{P}-\mathrm{B}\xb7\mathsf{\mu}}{\mathrm{a}\xb7\mathrm{B}}$ while if v = v

_{crit}, ${\mathrm{C}}_{\mathrm{v}}^{\mathrm{J}\mathsf{\sigma}}=2$. The positiveness of ${\mathrm{C}}_{\mathrm{v}}^{\mathrm{J}\mathsf{\sigma}}$ indicates that if the flow increases (due to a reduction in vaccination effort, see above) the generalized entropy generation, i.e., the negative impact over society, increases as well (and vice-versa). Notice also that ${\mathrm{C}}_{\mathrm{v}}^{\mathrm{J}\mathsf{\sigma}}\ge 2$ for all v, meaning that the vaccination effort causes a disproportional greater change in generalized entropy generation in relation to disease flow. However, in this case, the result is beneficial, since we are interested in decreasing the impact of the disease in society (i.e., we are in the light grey area of Figure 3). Figure 5 shows the plots of ${\mathrm{C}}_{\mathrm{v}}^{\mathrm{J}\mathsf{\sigma}}$ and of ${\mathrm{C}}_{\mathrm{v}}^{\mathrm{J}}$ as functions of vaccination effort.

_{a}can be performed for any of the parameters of the model, namely, P, a, μ, r, m, z. As previously mentioned, we do not intend to create a new model of vaccination, nor to analyze the model in terms of its equilibrium points and their respective stability or conditions to eliminate the contagious disease. Instead, our intention is to show an alternative way to interpret the impact of the parameters in the dynamics of the disease.

## 6. Conclusions

_{q}entropy to address eventual transitions between super additive and sub additive in the behavior of the system [43], might also prove relevant.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Cannon, W.B. Organization for Physiological Homeostasis. Physiol. Rev.
**1929**, 9, 399–431. [Google Scholar] - Billman, G.E. Homeostasis: The Underappreciated and Far Too Often Ignored Central Organizing Principle of Physiology. Front. Physiol.
**2020**, 11, 1–12. [Google Scholar] [CrossRef] - Goldstein, D.S. How Does Homeostasis Happen? Integrative Physiological, Systems Biological, and Evolutionary Perspectives. Am. J. Physiol. Regul. Integr. Comp. Physiol.
**2019**, 316, R301–R317. [Google Scholar] [CrossRef] - Windhorst, U. Regulatory Principles in Physiology. In Comprehensive Human Physiology; Springer: Berlin/Heidelberg, Germany, 1996; pp. 21–42. [Google Scholar]
- Nise, N. Control Systems Engineering, 3rd ed.; John Wiley & Sons, Inc.: New York, NY, USA, 2000. [Google Scholar]
- Levy, M.N.; Pappano, A.J. Cardiovascular Physiology, 9th ed.; Mosby-Elsevier: Philadelphia, PA, USA, 2007; ISBN 0323034462. [Google Scholar]
- Chaui-Berlinck, J.G.; Monteiro, L.H.A. Frank-Starling Mechanism and Short-Term Adjustment of Cardiac Flow. J. Exp. Biol.
**2017**, 220, 4391–4398. [Google Scholar] [CrossRef] - Solaro, R.J. Mechanisms of the Frank-Starling Law of the Heart: The Beat Goes On. Biophys. J.
**2007**, 93, 4095–4096. [Google Scholar] [CrossRef] - McEntire, K.D.; Gage, M.; Gawne, R.; Hadfield, M.G.; Hulshof, C.; Johnson, M.A.; Levesque, D.L.; Segura, J.; Pinter-Wollman, N. Understanding Drivers of Variation and Predicting Variability Across Levels of Biological Organization. Integr. Comp. Biol.
**2021**, 61, 2119–2131. [Google Scholar] [CrossRef] - Wade, M.J.; Kalisz, S. The Causes of Natural Selection. Evolution
**1990**, 44, 1947–1955. [Google Scholar] [CrossRef] - Tibell, L.A.E.; Harms, U. Biological Principles and Threshold Concepts for Understanding Natural Selection: Implications for Developing Visualizations as a Pedagogic Tool. Sci. Educ.
**2017**, 26, 953–973. [Google Scholar] [CrossRef] - Wade, M.J. Constraints on Sexual Selection. Science
**2012**, 338, 749–750. [Google Scholar] [CrossRef] - Barton, N.; Partridge, L. Limits to Natural Selection. BioEssays
**2000**, 22, 1075–1084. [Google Scholar] [CrossRef] - Saadat, N.P.; Nies, T.; Rousset, Y.; Ebenhö, O. Thermodynamic Limits and Optimality of Microbial Growth. Entropy
**2020**, 22, 277. [Google Scholar] [CrossRef] - Swanson, D.L.; McKechnie, A.E.; Vézina, F. How Low Can You Go? An Adaptive Energetic Framework for Interpreting Basal Metabolic Rate Variation in Endotherms. J. Comp. Physiol. B Biochem. Syst. Environ. Physiol.
**2017**, 187, 1039–1056. [Google Scholar] [CrossRef] - Heldmaier, G.; Ortmann, S.; Elvert, R. Natural Hypometabolism during Hibernation and Daily Torpor in Mammals. Respir. Physiol. Neurobiol.
**2004**, 141, 317–329. [Google Scholar] [CrossRef] - Ruf, T.; Geiser, F. Daily Torpor and Hibernation in Birds and Mammals. Biol. Rev.
**2015**, 90, 891–926. [Google Scholar] [CrossRef] - Boratyński, Z.; Koteja, P. Sexual and Natural Selection on Body Mass and Metabolic Rates in Free-Living Bank Voles. Funct. Ecol.
**2010**, 24, 1252–1261. [Google Scholar] [CrossRef] - Burton, T.; Killen, S.S.; Armstrong, J.D.; Metcalfe, N.B. What Causes Intraspecific Variation in Resting Metabolic Rate and What Are Its Ecological Consequences? Proc. R. Soc. B Biol. Sci.
**2011**, 278, 3465–3473. [Google Scholar] [CrossRef] - Rønning, B.; Moe, B.; Bech, C. Long-Term Repeatability Makes Basal Metabolic Rate a Likely Heritable Trait in the Zebra Finch Taeniopygia Guttata. J. Exp. Biol.
**2005**, 208, 4663–4669. [Google Scholar] [CrossRef] - White, C.R.; Kearney, M.R. Determinants of Inter-Specific Variation in Basal Metabolic Rate. J. Comp. Physiol. B
**2013**, 183, 1–26. [Google Scholar] [CrossRef] - Baškiera, S.; Gvoždík, L. Repeatability and Heritability of Resting Metabolic Rate in a Long-Lived Amphibian. Comp. Biochem. Physiol. Part A Mol. Integr. Physiol.
**2021**, 253, 110858. [Google Scholar] [CrossRef] - Ellegren, H. Comparative Genomics and the Study of Evolution by Natural Selection. Mol. Ecol.
**2008**, 17, 4586–4596. [Google Scholar] [CrossRef] - Bamshad, M.; Wooding, S.P. Signatures of Natural Selection in the Human Genome. Nat. Rev. Genet.
**2003**, 4, 99–111. [Google Scholar] [CrossRef] - de Groot, S.R.; Mazur, P. Non-Equilibrium Thermodynamics; Dover unab; Dover Publications: Mineola, TX, USA, 1984. [Google Scholar]
- Lucia, U.; Grazzini, G. The Second Law Today: Using Maximum-Minimum Entropy Generation. Entropy
**2015**, 17, 7786–7797. [Google Scholar] [CrossRef] - Lucia, U. Entropy Generation: Minimum inside and Maximum Outside. Phys. A Stat. Mech. Its Appl.
**2014**, 396, 61–65. [Google Scholar] [CrossRef] - Bejan, A. Fundamentals of Exergy Analysis, Entropy Generation Minimization, and the Generation of Flow Architecture. Int. J. Energy Res.
**2002**, 26, 545–565. [Google Scholar] [CrossRef] - Henriques, I.B.; Mady, C.E.K.; de Oliveira, S., Jr. Exergy Model of the Human Heart. Energy
**2016**, 117, 612–619. [Google Scholar] [CrossRef] - Yildiz, C.; Bilgin, V.A.; Yılmaz, B.; Özilgen, M. Organisms Live at Far-from-Equilibrium with Their Surroundings While Maintaining Homeostasis, Importing Exergy and Exporting Entropy. Int. J. Exergy
**2020**, 31, 287–301. [Google Scholar] [CrossRef] - Glansdorff, P.; Prigogine, I. Structure, Stabilité et Fluctuations; Masson & Cie Éditeurs: Paris, France, 1971. [Google Scholar]
- Chaui-Berlinck, J.G.; Bicudo, J.E.P.W. The Scaling of Blood Pressure and Volume. Foundations
**2021**, 1, 145–154. [Google Scholar] [CrossRef] - Lucia, U.; Sciubba, E. From Lotka to the Entropy Generation Approach. Phys. A Stat. Mech. its Appl.
**2013**, 392, 3634–3639. [Google Scholar] [CrossRef] - Visser, D.; Heijnen, J.J. The Mathematics of Metabolic Control Analysis Revisited. Metab. Eng.
**2002**, 4, 114–123. [Google Scholar] [CrossRef] - Reder, C. Metabolic Control Theory: A Structural Approach. J. Theor. Biol.
**1988**, 135, 175–201. [Google Scholar] [CrossRef] - Giersch, C. Control Analysis of Metabolic Networks. 1. Homogeneous Functions and the Summation Theorems for Control Coefficients. Eur. J. Biochem.
**1988**, 174, 509–513. [Google Scholar] [CrossRef] - Kacser, H.; Burns, J.A. The Control of Flux. Symp. Soc. Exp. Biol.
**1973**, 27, 65–104. [Google Scholar] - Kacser, H.; Burns, J.A. Molecular Democracy: Who Shares the Controls? Biochem. Soc. Trans.
**1979**, 7, 1149–1160. [Google Scholar] - Nogueira de Sá, P.G.; Chaui-Berlinck, J.G. A Thermodynamic-Based Approach to Model the Entry into Metabolic Depression by Mammals and Birds. J. Comp. Physiol. B
**2022**, 192, 593–610. [Google Scholar] [CrossRef] - Nogueira-de-Sá, P.G.; Bicudo, J.E.P.W.; Chaui-Berlinck, J.G. Energy and Time Optimization during Exit from Torpor in Vertebrate Endotherms. J. Comp. Physiol. B
**2023**, 193, 461–475. [Google Scholar] [CrossRef] - Batistela, C.M.; Correa, D.P.F.; Bueno, Á.M.; Piqueira, J.R.C. SIRSi-Vaccine Dynamical Model for the COVID-19 Pandemic. ISA Trans.
**2023**, 139, 391–405. [Google Scholar] [CrossRef] - Harari, G.S.; Monteiro, L.H.A. A Note on the Impact of a Behavioral Side-Effect of Vaccine Failure on the Spread of a Contagious Disease. Ecol. Complex.
**2021**, 46, 100929. [Google Scholar] [CrossRef] - StrzaŁka, D.; Grabowski, F. A Short Review of Elementary Properties and Possible Applications of Deformed Q-Algebra Derived from Non-Extensive Tsallis Entropy. Mod. Phys. Lett. B
**2008**, 22, 1525–1534. [Google Scholar] [CrossRef]

**Figure 1.**A general picture of levels of biological regulation/control. While the reductionist view would consider the upward processes at the left-hand side of the scheme as mandatory in controlling flows, the downward closed loops shown on the right-hand side create an entanglement between levels of biological organization that somehow precludes a simple cut-off of what is a controlled and what is a regulated variable (see, for instance, [9,10]). This is valid within the organism and among organisms as well since the biotic and abiotic environments would also dictate access to resources.

**Figure 2.**$\frac{\partial \mathrm{G}}{\partial {\mathrm{k}}_{\mathrm{a}}}$ axis as a graphical representation of the solutions of Inequations (16b) and (17c). The solutions pertaining to each inequation are indicated by cross-hatched bands. As it is shown, a simultaneous solution to both inequations is possible only if $\frac{\partial \mathrm{X}}{\partial {\mathrm{k}}_{\mathrm{a}}}<0$ and $\frac{\partial \mathrm{G}}{\partial {\mathrm{k}}_{\mathrm{a}}}>0$. The upper line is for the case $\frac{\partial \mathrm{X}}{\partial {\mathrm{k}}_{\mathrm{a}}}>0$ while the lower line is for the case $\frac{\partial \mathrm{X}}{\partial {\mathrm{k}}_{\mathrm{a}}}<0$. |·| indicates the absolute value.

**Figure 3.**The C

^{G}vs. C

^{X}plane. The elasticities’ coefficients of the conductance G and of the potential X are represented as orthogonal axes. The dashed line corresponds to C

^{G}= −C

^{X}and the dotted-dashed line corresponds to C

^{G}= −2C

^{X}. These two lines form the boundaries of the “best of both worlds” (bbw) condition of dJ > 0 along with dσ < 0, and of the “worst of two worlds” (wbw) condition of dJ < 0 along with dσ > 0. In the outside zones there are two grey areas, one indicating the region with dJ > 0 and dσ > 0 (dark grey) and the other indicating the region with dJ < 0 and dσ < 0 (light grey). Both these regions have non-negative ${\mathrm{C}}^{\mathrm{J}\mathsf{\sigma}}$. However, in the former (dark grey zone), processes that occur in the 2nd quadrant are the most efficient, and processes that occur in the 4th quadrant are the most inefficient, while the opposite occurs in the latter (light grey zone). The 1st and the 3rd quadrants are of intermediate efficiencies in their respective zones. Numbers in parentheses indicate the ${\mathrm{C}}^{\mathrm{J}\mathsf{\sigma}}$ value over the respective line. For instance, along the C

^{X}axis, ${\mathrm{C}}^{\mathrm{J}\mathsf{\sigma}}=2$, while along the C

^{G}axis, ${\mathrm{C}}^{\mathrm{J}\mathsf{\sigma}}=1$. These results come immediately from Equation (15) since walking over one of these axes means to have no changes in the other.

**Figure 5.**${\mathrm{C}}_{\mathrm{v}}^{\mathrm{J}\mathsf{\sigma}}$ (red) and ${\mathrm{C}}_{\mathrm{v}}^{\mathrm{J}}$ (blue) versus vaccination effort. See text for detailed discussion.

Parameter | Dimension |
---|---|

P | [n t^{−1}] |

a | [n^{−1} t^{−1}] |

B | [t^{−1}] |

v | [t^{−1}] |

z | [n^{−1}] |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bicudo, J.E.P.W.; Nogueira-de-Sá, P.G.; Chaui-Berlinck, J.G.
Entropy Generation and Control: Criteria to Calculate Flow Optimization in Biological Systems. *Foundations* **2023**, *3*, 406-418.
https://doi.org/10.3390/foundations3030029

**AMA Style**

Bicudo JEPW, Nogueira-de-Sá PG, Chaui-Berlinck JG.
Entropy Generation and Control: Criteria to Calculate Flow Optimization in Biological Systems. *Foundations*. 2023; 3(3):406-418.
https://doi.org/10.3390/foundations3030029

**Chicago/Turabian Style**

Bicudo, José Eduardo Pereira Wilken, Pedro Góes Nogueira-de-Sá, and José Guilherme Chaui-Berlinck.
2023. "Entropy Generation and Control: Criteria to Calculate Flow Optimization in Biological Systems" *Foundations* 3, no. 3: 406-418.
https://doi.org/10.3390/foundations3030029