Yuri K. Shestopaloff


Physics of Growth and Replication


Physical and geometrical perspectives on living organisms’ development



Copyright © 2010 by Yuri K. Shestopaloff.


All rights reserved. Permission is granted to copy both the textual and graphical excerpts from the book for educational, professional, academic or other non-commercial use, with a clear indication of the publication data (the author, book title, publisher, place of publication, year of publication and edition), and copyright notice.


ISBN 978-0-9809667-5-6  

Library of Congress Control Number:  2009943094


Published by AKVY Press

Toronto, Canada

Coral Springs, Florida, USA


Library and Archives Canada Cataloguing in Publication


Shestopaloff, Yuri K., 1955-
        Physics of growth and replication : physical and geometrical perspectives on living organisms' development / Yuri K. Shestopaloff.


Includes bibliographical references and index.
ISBN 978-0-9809667-5-6


        1. Cells--Growth--Mathematical models.  2. Cell proliferation—Mathematical models.  3. Cells--Mechanical properties--Mathematical models.  4. Cell division--Mathematical models.  5. Growth--Regulation--Mathematical models.  I. Title.


QH511.S55 2010                     571.8'49                     C2010-900110-9



This book introduces the physical mechanisms that besides the biochemical and other factors govern the growth and replication of living organisms. Presently, these phenomena are overwhelmingly considered as biological, genetic, and biochemical, which is true. However, natural phenomena exist in a real world that does not have purely physical or biological boundaries, which subjectively imposed by our excessive human desire for classification and absolutization. Physical laws, as an inherent part of Nature, influence such universal phenomena as growth and replication of living organisms. The book presents detailed study of such fundamental physical growth mechanism. It introduces a general mathematical equation describing the growth of living organisms from the physical and geometrical perspective, and applies it to modeling of different growth scenarios supported by experimental observations. Cells’ growth is the primary area of the study, although the author also makes some generalizations and comments on related natural philosophy issues.  One small chapter considers the growth of multicellular organisms. The whole material is presented in such a way that any person interested in the subject and familiar with the basics of physics and biology will be comfortable with the main concepts and the whole content, which is much facilitated by numerous graphical illustrations. 


"Science only for scientists" is as much nonsense

as "literacy only for nobles"


                                            Yuri K. Shestopaloff




Table of Contents


 TOC \o "1-3" \f \u Introduction..............................................

Acknowledgements and other thoughts       

About the author....................................


1. The present view of the growth and replication phenomenon from the physical perspective...........................

1.1. Summary of overview..................................

2. Cells’ growth and replication.......

2.1. Growth and interaction with the environment     

2.2. Growth of nuclei............................................

2.3. Replication resolves the cell’s conflicts.

3. Mathematical presentation of the surface-volume growth mechanism 

3.1. Introducing the notion of relative growth

3.2. Deriving the surface – volume growth equation    

3.3. Modeling the growth of cells with constant membrane’s inflow

3.4. Growth of cells with varying membrane’s inflow   

3.5. Amoeba growth..............................................

3.6. Cells’ shape as a growth suppressing factor

3.7. Growth of embryonic cells in Drosophila, and the growth-volume suppression mechanism..........................................................................

3.8. Cells’ elongation during the growth of blastocysts in pigs         

4. Other growth scenarios. Growth and replication of E. coli   

4.1. Changes in membrane’s inflow of E. coli

4.2. Growth of a spherical cell with varying membrane’s inflow    

4.3. Changing the growth stopper value as an alternative to fit the growth curves   

4.4. Growth of elongated cells with varying membrane’s inflow       

5. Growth ratios. Cells’ overgrowth through the form change    

5.1. Variations in correlation between the elongated shape and the growth rate     

5.2. Growth ratio conjecture............................

5.3. Overgrowth and natural selection.........

5.4. Surface – volume mechanism and cells’ life cycle  

6. Growth of organisms and systems

6.1. Growth equation for multicellular organisms and organs      

6.2. Generalization of the surface-volume growth mechanism      

6.3. Growth mechanism and evolutionary development         

6.4. Workings of the surface – volume mechanism on organism level     

7. Conclusion........................................


Index Table...........................................




The material presented in this book has been in part exposed in the previous publications. The comments and reviews have been thoroughly analyzed and certainly have been incorporated into this book in one way or another. Overall, we significantly advanced the subject compared to the previously published material. First of all, the area of experimental observations, to which the growth equation has been applied, became wider. For instance, beside Amoeba, we also consider E. coli bacteria, Schizosaccharomyces pombe and model its growth.

Comparison to available experimental data showed that the growth equation describes the growth of these bacteria very well, including slight convexity of the graph, which serves as a battle field for many debates over which function better approximates the cell growth – linear or exponential. In fact, the issue is not as straightforward (and it cannot be, given the diversity of life forms and their environments, and accordingly growth scenarios). In some instances, as it is the case of Amoeba growth, the growth of mass on time is described by concave curve (bent downward), sometimes the growth dependence is very close to linear. In some scenarios, as it is the case of E. coli, Schizosaccharomyces pombe, the growth is approximated by convex function (bent upward). In the last case, the exponential curve fits experimental data better than the linear function, although this is not necessarily the consequence of really exponential nature of the growth phenomena, but rather a matter of choice of statistical method. The curvature is small, and this is why very wide class of functions, for instance, polynomial, can be used for the same purpose.

These debated growth scenarios found a well grounded explanation within the concept of the physical growth mechanism. In fact, all mentioned growth scenarios, and many more, fit perfectly to the growth framework provided by the physical growth mechanism. One of the major differences in growth scenarios is this. All growth curves (describing dependence of the growing mass on time) follow the growth curve defined by the physical growth mechanism. However, depending on a particular environment and evolutionary pressures, some organisms use only the fastest part of the growth curve and then switch to replication. This is an efficient solution for organisms that need to grow and replicate fast in order to survive. Other organisms, which live in a stable environment and that do not need replicate fast, use almost the whole growth cycle defined by the growth curve. Amoeba is an example of such an organism.

Of course, this is not the only new material added to this book. In fact, the level of research has reached the point where the introduced growth formula became a working tool that can be used for practical purposes. In particular, using the growth formula, we discovered the increase of nutrient supply through the unit of the membrane’s surface during the growth. Moreover, it is possible to quantitatively evaluate this increase, that is to find the dependence of nutrient supply through the unit surface of the cell’s membrane on the size of the cell, or, in other words, its growth phase.

Overall, this stage of study of the physical growth mechanism, presented in the book, showed that this mechanism plays more prominent role than we thought before, being responsible for the growth effects of the “second” order, such as, for instance, the convex form of the growth curve. On the other hand, the same fact is a remarkable demonstration how closely biochemical, biological and physical growth mechanisms interconnect; all of them work together and confluence each other, while tirelessly creating a single unity – the cell or the whole organism.




I am thankful to my family for the inception and support of the entire project. Some people, during the course of this research, helped me in one way or another. I am grateful to Dr. Alfred Tauber and A.N. Schechter, who reviewed the earliest versions of the material and provided useful feedback. I would like to say the greatest thanks to Dr. Piotr H. Pawlowski from the Institute of Biochemistry and Biophysics, in Warszawa, who reviewed the previously published book and provided encouraging feedback and support to continue this research despite many impeding factors.



Prehistory of the research and other thoughts


A while ago, I wrote a short article, just several pages, about the fundamental mechanism responsible for the cells’ division, considering this phenomenon as an example of workings of general (dialectical) laws. The thing appeared to me so obvious, that I had been almost sure that somebody else already came with such an idea, although most likely in a different way, while my starting point was a purely philosophical dialectical consideration about interaction of dialectical opposites presenting in any Nature’s phenomenon. (This law is one of the fundamental reasons of all changes we observe around us.)

The article was supposed to be a forever forgotten paper, as many others writings; it was destined to oblivion. (To be fair, this happens not because of the writings’ quality, but because of my reluctance to market them.) This has been my son who pulled this article out of curiosity from the pile of papers in the basement and brought it to me, asking questions what this paper was about. I explained, and then hesitated for several weeks, thinking whether to take any further steps. As you can see, I eventually did.

It took time to derive the growth equation. Its introduction and proof of validity is the core subject in this book. (Adequate quantitative model of any phenomenon is very important for its understanding and application, beside the thorough and comprehensive perception in all other knowledge dimensions.)

The present situation is that the hypothesis, although  known and to some extent appreciated, did not receive enough attention of the whole community to make its acceptance or at least awareness about it irreversible. So, I have to continue my efforts trying to save it from disappearance. It still can happen, and the history presents lots of evidence how many good and outstanding inventions and findings vanished without a trace, and this discovery can easily follow the same fate. (Here, I implicitly assume that the growth hypothesis is valid, but in fact this is still a hypothesis, although with very solid proofs at this point.) It took more than two thousand years to rediscover the alloyed steel used by Carthaginians. However, this is a drop in the forever dried sea of disappeared discoveries and inventions, which we will never know about. In fact, this civilization as a whole has way too much hubris to the past and overvalues its achievements so much that it becomes more and more ignorant of laws governing its existence as part of Nature. Such inadequacy of perception of the reality has a surprisingly costly price, which eventually has to be paid. The further this payment is deterred, the more “interest” will accrue. This is a material world in all meanings of this word, and there is no way to avoid this payment. But that’s us, humans.



About the author


Yuri K. Shestopaloff is a professional academic and consultant. He started his academic career as an Applied Mathematician and Engineer-Physicist. He received his M.Sc and Ph.D degrees from Moscow Physical Technological Institute, focusing on the development of mathematical methods and algorithms for interpretation of remote sensing data. He received the Doctor of Sciences degree, the highest academic degree in European countries, for developing mathematical methods for data interpretation and processing. He has worked as Associate Professor, Full Professor and Chair at the Electrical Engineering Department of Academy of Transport, and held position of Chief Scientist at Institute of Sensor Microelectronics.

Yuri does consulting and research on mathematical modeling and development of mathematical methods, computational algorithms and data processing in various fields of science and technology, such as financial mathematics, biology, remote sensing and wave propagation theory. He also consults on system design issues, middle and large scale software development projects. He is regularly invited to speak at professional forums and conferences on financial mathematics and mathematical modeling. Yuri has published six books and over eighty academic articles. His literary writings (poetry, short stories and novels) are appreciated by a wide audience. Yuri publishes also articles on philosophical issues, mostly on natural philosophy and acquisition and validation of scientific knowledge.




Many scientists share the opinion that the form of biological objects influences their growth and replication. This concept has been suggested early by Aristotle and has many supporters, among which D’Arcy Thompson Wentworth developed it in depth and advanced to other physical phenomena beside the growth. This opinion have had many followers in the previous decades. However, no complete and really convincing studies have been done, and no quantitative models have been developed to firmly support this hypothesis XE "growth hypothesis" . As a result of such development, and given the impressive achievements in the molecular biology in the last decades, this camp weakened and eventually abandoned their trenches. In the recent decades, the staggering successes of genetics, cellular and molecular biology discovered so many particular biochemical mechanisms directly or meditatively affecting the growth and replication of cells and organisms, that the overall perception of this complex phenomenon overwhelmingly swung to the opposite side rejecting any other growth factors except purely biological and biochemical. This is quite normal situation with us, humans, exploring the course of life jumping from one opposite to another, and the author of this book is by no means an exception from this rule. In fact, the Nature, and accordingly the scientific knowledge, are governed by measure[1], this is where the optimum usually resides.

There is another consideration that enforces this position that considers natural phenomena as multifactor ones, and which has many supporters. Natural phenomena exist in Nature continuously, they are not (and cannot be principally because of this continuity) separated from all other natural factors that potentially influence some phenomenon. The influence of these factors can be really small, but it must present anyway because of the continuity of matter and consequently the continuity of laws of Nature governing the evolvement of this matter.

With regard to the growth processes, the physical factors cannot be even small because of their omnipresence in this range of dimensions, and universality of the growth.  So, the growth of living organisms principally has to be a multifactor phenomenon. The issue is to find what factors influence, to what extent, and how. To do this, we need quantitative apparatuses adequately modeling the growth processes. The growth should be considered as an entity belonging to the whole realm of Nature, which does not have these strict boundaries as we, humans, are often trying to impose to Nature, but includes all factors from different areas, known and unknown to humans, in their harmonic and compatible coexistence. In the previous works, this list, “harmonic and compatible”, also included the word “optimal”. However, the optimality assumes certain criteria, and this is why, if we want to use this notion in the above statement, we would have to embark into a long discussion about criteria of an optimal development, which we do not need in this book. However, the notion of optimality remains true with regard to organization of collective influence and interaction of many factors.

In this book, we study further the hypothesis that the discovered growth mechanism, that is based on the interaction and unity of surface and volume of a living organism, presents a fundamental physical mechanism that defines the growth and replication. Of course, this physical mechanism is not alone player in such a complex phenomenon as growth, but presents together with the genetic, epigenetic and biochemical mechanisms, and surely many others we are not yet aware about. First, we derive a general mathematical equation describing the growth from the physical and geometrical perspectives. Then, we apply it to modeling of cells and their growth. We compare the computed results to experimental data.  Later, we generalize this physical growth mechanism for other biological objects such as organs and multicellular organisms.

The validity of the introduced growth mechanism, and associated with it growth equation, is well supported by experimental results on Amoeba growth XE "Amoeba growth" , development of trophectodermal cells XE "trophectodermal cells"  in pigs’ blastocysts XE "blastocysts" , cellularization of the syncytial blastoderm XE "syncytial blastoderm"  of Drosophila, growth of E. coli bacteria, existence of giant cells and giant nuclei, etc. We discovered several new growth related effects using the growth equation as a modeling tool for the cells’ growth. One of them is finding of a new growth suppression mechanism XE "growth suppression mechanism"  based on the cells’ geometry. The other one is the increase of nutrients flow through the unit of membrane’s surface at a unit of time during the growth process. Although this fact can be deduced from the geometrical considerations, the application of growth equation allows finding quantitative parameters and the appropriate functional dependences describing this increase.

The importance of the geometrical approach to the growth phenomenon received another important consideration. In fact, the growth equation introduces the characterization of geometrical forms that can grow. We cannot have the living creatures or organs of an arbitrary geometrical form. There are strict criteria and limitations on that imposed by the physical growth mechanism (A. Shestopaloff)[2].

This book solves another controversial issue debated now for more than half a century. This is the problem, which curve, linear or exponential, better describes the growth process. This is a common sense that there is no definitive answer to such a question, and cannot be, because so many living forms and such a great diversity of environments exist, while the growth is a universal phenomenon governed by general laws. If one thinks for a moment, he will realize that these laws cannot be reduced to such simple description as only linear or only exponential dependence of growth of the cell’s mass on time. This is Nature, it is much more elegant creator than such simple assumptions could imply. It is not complicated, its functioning is based on relatively few number of fundamental laws of different levels of generality. Such universality excludes complex arrangements and mechanisms. Complex things cannot be universal, not to say adequate to the reality in the wide range of parameters, while the general laws have to describe the infinite number of possible scenarios. This is why the general laws include relatively few core parameters, so that the rest of particularities can be derived in each particular case. However, the general laws are not trivial either, their complexity is exactly adequate to their universality, not more, nor less. This is where the dialectical category measure is doing the finest work possible, producing the polished and refined by the eternity and motion of matter general laws of Nature.

The introduced growth mechanism is an example of such an ingenious craft of Nature. This is a general law. It works universally forming all living organisms regardless their size. The other important feature of the growth mechanism is that it works on all levels of organism simultaneously, shaping cellular organelles, the cell, organs, organism as a whole and its systems, all at once. In fact, the mere origin of organs is also the result of working of this growth mechanism redistributing volumes and surfaces in such a way that provides functioning of the whole organism. The law of compartmentalization, which we will introduce and describe briefly too in this book, works together with the growth mechanism (and others as well), so that many factors and mechanisms influence the growth. We can think, for better understanding, that the physical growth mechanism acts, to some extent, as a placeholder, within which the biochemistry growth and replication machineries deploy, as well as other types of associated processes and mechanisms. On the other hand, all these factors influence the workings of the physical growth mechanism as well. The unity of all growth mechanisms is an ensemble playing a single “growth symphony”.

We think that the results we present in the book eventually might have the diverse theoretical and practical implications in biology and medicine, provided this discovery will not disappear because of the author’s inability to convince people in its validity and deliver the reachable message. (Well, in this case, the author is not the only person to blame!) Most likely, despite the author’s efforts and optimism, the results are not going to be recognized soon, given the current situation in science and in modern societies in general experiencing some deficit of a broad and unobstructed by specialization vision.  It does not mean that there are no such people, not at all. There are many of them, but the opponents are numerous, the environment is favorable for them, and this is why it is difficult to overcome the resistance. On the other hand, this is a normal thing, meaning the existence and interaction of the dialectical opposites. Any action creates a counteraction. This is not a mechanically understood arrangement, the matter of dialectical opposites is more subtle, but it is always there, and the case of the physical growth mechanism, namely the problem of its recognition, is not an exception.

Today, the pendulum of the current public opinion has a high momentum on the side of molecular biology. It will take a while when it begins to move back under the weight of many facts and phenomena unexplained from the purely biochemical perspective. Maybe then, this material will get more attention if not forgotten entirely at that time. Anyway, we decided to throw the seeds of this knowledge into the presently unfriendly and dry soil in a hope that some of them will miraculously survive and will grow up some day; in the same way, as the seeds from the old sack grew up when Robinson Caruso XE "Robinson Caruso"  just emptied it. (We still believe in stories and miracles, even if we have grown up a long time ago; but that’s our human nature!) This is the beauty and the danger of this world that much can be changed due to directed efforts, and this book is the part of a plan supposed to make the recognition of this fundamental growth mechanism to happen.

The book may not convey enough proofs to convince strong adherents of a purely biological growth paradigm. (Logic can reshape beliefs as much as a toothpick can change a diamond. The logical proofs are for knowledge, not for beliefs.)

We think, an attentive and thoughtful reader will find enough worthwhile points and sound considerations in the book. We expect that a more elaborate and voluminous book will follow, which hopefully will compensate for the brevity of this one, and fill the canyons of suspicion to make the whole landscape of these conceptual approach more friendly, and potentially habitable. (Eventually, Romans came to the valley that was previously uninhabited because of the eruptions and earthquakes in the previous centuries. And then, starting from a small tribe, they changed and shaped the world so much. For good or bad, this is another question, but they certainly did.)



[1] On the other hand, the saying “Everything in measure including measure” is also true. However, for complex multifactor phenomena (and the growth is certainly such a phenomenon), the norm is the measure, not the boundary or extreme solution.

[2] The idea of characterization of geometrical forms that can grow has been suggested by Alexander Shestopaloff during the discussions of the physical growth mechanism and its conceptual geometrical content.

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Last modified: 01/10/21