Cellular growth, injury and death
0 Pioneering Research Division , U. S. Army Natick Laboratories, Natick, Mass. , U. S. A
Mathematical formulation and analog computer analysis KURZFASSUNG: Zellwachstum, Sch~idigungund Tod. Mathematische Formulierung und Analog-Computer-Analyse. Vom Gesichtspunkt molekularer Mechanismen sind biologische Vorgi/nge auf~erordentlich komplex. In der vorliegenden Studie wird der Versuch unternommen, auf der Basis eines Modellsystems das Zusammenwirken funktioneller Grundeinheiten der ZeIle zu einem System aufzulaellen. Die Fluflglei&ungen des Modellsystems werden aufgestellt und in die entspre&enden Differenrialglei&ungen trans?ormiert. Fiir den Wachstumsvorgang wird unter vers&iedenen Parameterbedingungen eine Analog-Computer-Analyse dur&gefiihrt; ferner werden die Ph~inomene Zellsch~idigung und Tod analysiert. Experimentelle Studien am Computer gewi/hren wichtige Einbli&e in die grundlegenden Wachstumsvorgiinge und in die Wa&stumsdesorganisation dur& iiuflere Einwirkungen.
I N T R O D U C T I O N
Growth is a basic property of biological species, and growth coupled with the
division process leads to the increase of population. Cellular growth can be
considered from various points of view, for example, as it exists at the present time,
thus ignoring evolutionary aspects, since evolution represents the development of new
biological systems and new biological properties. The complexity of biological systems
is obvious. It is not so obvious how to gain more insight into detailed mechanisms
inherent in cell growth and integrated process. Collecting data from isolated
experiments yields important raw material for the synthesis of more complex systems. H o w
ever, detailed information obtained from experiments performed in conditions which
are not equivalent to cellular environment represents only a limited type information,
and such information has to be integrated into a larger framework of cellular
operations. Current analysis of cellular growth processes in the literature is essentially
carried out by a descriptive process. However, verbal nonquantitative analysis of
nonlinear systems is not adequate. The usual intellectual reasoning process involves only
a few steps of linear logic. Biological processes are essentially non-linear. Furthermore,
they represent a multiple network of interactions and, consequently, application of
linearity into such framework is leading to an illusion of understanding rather than a
real understanding of the basic process. Since biological processes when considered at
molecular level involve a large number of functional entities, the descriptive analysis
of such a system has only a limited value.
In order to comprehend the basic cellular processes, it is essential that
relationships between individual functional units should be analyzed as a function of time,
since the growth process represents the change of absolute concentration of those
entities. Therefore, it is essential that formalized relationship be established between
the operational elements of the system and that the problem be solved in a quantitative
manner. In order to do this it is imperative that a model-system be established. In
cellular metabolic systems there are a number of similar-type elementary units which
are organized into complex patterns. It is considered that an analysis of a
modelsystem which contains the principal functional entities, such as DNA, RNA, and
enzymes, may lead to an understanding of growth processes and their regulation.
The development and construction of a m o d e 1- s y s t e m is not at all an easy
process. It would be desirable to have a very complete model, which would help
interpret many complex cellular processes. However, quantitative formulation is limited
by many prohibitive factors. Consequently, the complexity of the model is limited to
that level where the mathematical and computer techniques permit it to be carried
out. Our model-system contains all basic functional entities which are operational in
the cell, and they are organized in a framework of mutual interaction as known from
the data in the literature.
In general, the basic elements of the system are: genes, messengers (RNA),
templates, enzymes, repressors and activators. The model-system contains four basic
functional genes which represent group-properties. This approach is justified if one considers
that group-property is based on the fact that all basic functional entities are built up
with a certain number of integral building blocks, and a complete number of blocks
have to be there in order to build a basic functional entity. If one building block is
missing, a functional unit cannot be built. Consequently, group-property will give a
cross-behavior of the system. In the initial condition it is assumed that all functional
entities have a relative value of ul~ity, and as a function of time all entities will start
to grow. They reach approximately double value at the end of the generation time,
which has been selected as a convenient time interval in the framework of computer
observation time. The growth of the model-system is initiated from initial conditions
by activating the external pool. This is equivalent to taking cells and placing them
in a nutrient medium and measuring their growth.
FORMULATION AND DESCRIPTION OF MODEL-SYSTEM
The terminology used in model building is represented in Table 1 and the flow
schemes in Table 2. A schematic diagram of the model is presented in Figure 1.
Let us review briefly the operational characteristics of a model-system. The first
functional process can be considered the complexing of messenger M with ribosome B,
yielding the template N. Messenger M is produced by interaction of complex [Ep Pn]
with gene GE. Enzyme synthesis results when templates interact with transport RNA
and the amino acid pool complex [C Pal. The total protein E contains three fractions:
En - enzymes which convert internal pool Pi into RNA percursors; E~ - enzymes
which convert internal pool Pi into amino acids; E~ - enzymes which convert external
pool P~ into internal pool Pi. Rate constants ( k l , ka, and kt) determine what fraction
of the total protein is converted to respective enzymes. The formation of polymerase
Ep follows generally the same line represented by total protein E synthesis. External
E - Total protein
E n - Enzymes which convert internal
pool (Pi) into RNA precursors
E a - Enzymes which convert internal
pool (Pi) into amino acids
E p - RNA polymerase for messenger
RNA (M) synthesis
E~ - Enzymes which convert external
pool (Pc) into internat pooi (Pi)
Rate constant kn, ka, and kt determine
what fraction of total protein represents
GE - Genes for messenger RNA (M)
Gp - Genes for messenger RNA (Mp) synthesis G B - Genes for the synthesis of RNA fraction of ribosome
Go - Genes for transport RNA (C)
M - Messenger (RNA) for protein (E)
Mp - Messenger (RNA) for Ep synthesis
B" RNA fraction of ribosome
B - Ribosome
C - Transport RNA
N - Ribosome and messenger complex
for protein (E) synthesis (template)
N p - Ribosome and messenger complex
for Ep synthesis (template)
N* - Inactive state of N
N ' p - Inactive state of N~
si - M e t a b o l i c which converts
plates N and N p into inactive state
s'i - Metabolic which converts inactive
template N* and N*~ into active
kl...k, - Various rate constants
nutrient pool Pe is converted into an internal pool Pi. In addition, p a r t of the pool Pi
leaks out from the cell via kao, but the product X will not be associated with the
external pool. The principal regulatory element in the system is the complexing of total
protein with the internal pool (Eq. 22, Table 2). This complexing is reversible and the
ratio between the two rate constants, k~2, k.-~2, will determine the degree of regulation
at that level. Additional regulatory features are the complexing of polymerase with
transport RNA (Eq. 21) and the complexing of polymerase with ribosomes (Eq. 23).
All functional entities have different degrees of stability, and they decay into their
respective pools. The system is growing because there is an input via transport enzyme
Et from the external pool. In order to analyze cellular dormancy, two additional
entities are introduced into the system. It is visualized that template N can be
converted into inactive form N* and template Np into N*p. This conversion can result
from the action of metabolites or hormones and is considered to be reversible by
In order to analyze this type of model-system in a quantitative manner, it is
essential that a systematic formulation be made. Consequently, the flow scheme in
Table 2 is organized into a set of simultaneous differential equations, and these are
programmed for the analog computer. This procedure is not further discussed here,
since this has been analyzed in previous publications
(HI~INMETS1964a, 1964b, 1966)
When the program has been assembled on the analog computer, it is essential to
establish a f u n c t i o n a 1 s y s t e m. The principal characteristics of a functional system
for the cell growth can be considered to be the requirement that during the generation
time all cell components have to double their initial value. Organization of this
requires a great deal of patience and experience in computer technology since the
model contains many entities (31 rate constants). Only ai?cer tedious and prolonged
trials is it possible to develop a functional system. The principal cause of the difficulty
is the fact that the system is highly nonlinear. As a matter of fact, it is a network of
nonlinearities. This is of course the principal feature of the biological system, where
many degrees of interdependence exist among functional entities.
Once the functional system has been established on the computer, then it is
possible to carry out experiments on the model-system and observe kinetic behavior of the
system on an oscilloscope screen. It is instructive to observe the results when certain
rate constants of the system are changed or the initial conditions altered. Varieties of
experiments of this nature can be carried out and subsequently recorded for permanent
records. Thus laboratory experiments can be carried out on the computer. For example,
many biological experiments are using techniques in which a certain compound is
introduced into the system and perhaps removed or neutralized later. In order to carry
out experiments of this nature, various electronic switches were incorporated in the
program of the model-system. They were synchronized with computer solutions so
that operations could be carried out at any desired time. Also, experiments of a more
complex nature were designed so that two compounds could be simultaneously
introduced into the system and simultaneously removed from the system. The transient
introduction of various elements into the model-system enables one not only to follow
the events at the site where the interaction takes place, but also to follow a sequence
of events which take place throughout the system. This is in contrast to experimental
procedures in biology where very few entities can be simultaneously studied because
of limitations of experimental techniques. Consequently, the amount of information
obtained from laboratory experiments can be more iimited so far as the total system
is concerned. This is especially true for the kinetic aspects of the process. On all
recordings the initial value for the functional entity is indicated by 1, and 0 indicates
that the functional entity has disappeared from the system. Value 2 indicates that the
particular entity has doubled.
ANALYSIS OF G R O W T H PROCESS AS A F U N C T I O N OF TIME
A computer solution is obtained for the cell growth when the model-system is
made operational in the presence of external pool Pe. Since all basic functional entities
have the initial value of unity, then the growth will be initiated from that level. In
order for the system to grow, it is essential that there should be an in-flow of pool Pe
via transport enzymes Et. Rate constant k~8 value has been so selected that during the
generation time all cell components approximately double the initial value. The growth
of the model-system when it starts from the initial values is in an arbitrary state. In
the initial phase there are formations o? various complexes and adjustment of
equilibrium at various steps. Only afLer the initial transient phase has passed can we
consider the normal growth conditions established and the system to be suitable for kinetic
analysis. In order to minimize the initial transients, one has to assign initial values for
complexes (see Eqs. 1, 9, 20, 21, 22, Table 2). These were determined empirically for
conditions where there are only small initial transients present. Figure 2 shows the
growth of three pools, Pi, P~, and Pn- Pool Pi increases rapidly up to a maximum value
and then decreases gradually. When observation time is extended beyond generation
time, this curve wilt flatten out and gradually start to rise again. Pool P~ aider the
initial transient will also start to grow, finally reaching a plateau. It is obvious that
the amino acid pool and general internal pool have quite different growth
characteristics. In contrast, the nucleotide pool P~ will pass through a steep reduction phase
before it will start to grow. This of course can be expected since the first step of
synthesis will be utilization of Pn by genes, and this causes a rapid initial loss of this pool.
In general, processes are so complex that no attempt is to be made here to analyze the
curve forms specifically. The formation of messenger M, template N, enzyme E, and
ribosome B all have different kinetic characteristics (not presented here). However,
finally all entities start to grow and reach approximately the same value at the same
generation time. Obviously this system would grow and explode if cellular division
did not take place. Since we are considering at the present only growth, no attempt is
to be made to analyze the growth beyond generation time.
T H E EFFECTS OF INTERNAL POOLS O N T H E G R O W T H
CHARACTERISTICS OF THE FUNCTIONAL ENTITIES
At first we consider the rate effects arising from the conversion of pool Pi into
amino acid pool P~, which is accomplished via rate constant k.2o. Figure 3 shows the
enzyme E~ growth at various k20 values. It is evident that higher k20 values produce
a more rapid growth and low values terminate the growth. Other entities (not shown
here) exhibit in general the same growth trends.
In order to study the effect of nucleotide pool formation, the rate constant k19
is varied, while the growth of functional entities is recorded. This effect is
demonstrated in Figure 4, where transport RNA C growth is presented at various k~9 values.
It is evident that at a low k19 value ("3") there is after small initial rise a rapid
reduction of C concentration. This is followed by a leveling off phase, and finally an
extremely slow growth is established. At higher kl9 values, the final growth becomes
progressively larger, but growth of C will be suppressed at very high kl9 values ("15").
This experiment shows clearly that it is difficult to predict by casual reasoning how a
complex system behaves at various parametric values. Other entities have been
recorded in similar conditions, but are not presented here. The general conclusion which
is drawn on the basis of experimental data reveals that excessive values of k19, either
too large or too small, will produce abnormal concentration relations between
functional entities. As a consequence, abnormal growth characteristics develop in the
The effect of a sudden reduction of pool Pa on the growth of various functional
entities is shown in Figure 5. It is evident that all entities exhibit different kinetic
characteristics following such a transient event. Enzymes E and Ep start to decline
while ribosome B, after the initial decline, starts to grow rapidly. All other entities
suffer concentration changes in various degrees. However, if computer observation time
is extended to the range of several "generation times", it is evident that finally alt
functional entities start to decline, and system becomes extinct. I t appears that the
model-system becomes non-operationaI when excessive changes are made in single
parameter values. The significance of this event in terms of biological phenomenon
will be evident when cellular injury mechanisms are studied.
CELLULAR I N J U R Y A N D D E A T H
I n t r o d u c t i o n
The phenomenon of cellular injury and death is a very perplexing problem in
nature and has many ramifications. I t has been a subject of many speculations and
many theories have been proposed to interpret the mechanism of cellular death.
Before we analyze this problem in more detail, it is important to consider the viability
problem in general and explore its practical aspects. I t is essential to put the concept
of cellular viability on a rational basis, and one should attempt to interpret cellular
injury and death via basic metabolic and synthetic processes. Furthermore, one cannot
consider cellular death only from the point of view of complete and absolute killing.
There are degrees of injury, and consequently there may be cellular recovery. In
principal, there is no absolute concept of cell life or death. There are states of functional
disorganization, and those states can be altered by internal or external factors.
In general, a cell can be killed or injured in a multitude of ways. It can be killed
by radiation, by excessive temperatures, by mechanical injury, by chemical agents, etc.
This raises the question of whether the process of death is basically a singular process
or is a diverse and complex set of disintegrative phenomena. It would be of benefit to
review briefly the basic views existing at the present time on this subject in the
literature. Many theories have been proposed to interpret the cellular killing process, but
of foremost importance from the view of popularity is the so-called target theory. It
is based essentially on population statistics where cellular inactivation or killing rate
will provide information in regard to the killing process which may be a single hit or
(DAvis & F~INOOLD 1962, L~A 1947, RAHN 1945)
. Despite its obvious
shortcomings, the target theory is still widely accepted as a means of interpreting and
elucidating the mechanisms of cellular death (GRAY 1961,
MAI~covmH 196I, HAYNES 1964). Since target theory is based on the curve form analysis,
which itself is affected by a multitude of experimental parameters
(WrBB 1964, Power,s
, one cannot expect that the "death kinetics" of a population will yield
information with regard to the cellular killing process and mechanism of cellular recovery
on a single cell basis. In the literature numerous curves have been published for such
analysis. No significant results have been obtained. One could ask whether the theory
is founded on the premise which potentially could give information for cellular death
mechanism, or are we dealing here with an obscure illusion? The primary source of
information for target theory is the relationship between the rate of killing and the
exposure dose. In our opinion, this type of relationship does not contain any basic
information which is necessary for interpreting cellular killing mechanisms.
A few decades ago it became obvious to some that target theory was not capable
of explaining cellular injury nor cetlular death, but no substitute for the theory
appeared for a while. Then some new proposals were made by us for the
interpretation of cellular injury and death phenomena (H~INMrTS 1954). Later, a wider
and more comprehensive treatment of the subject was made
basic premise proposed was based on the concept that there is a definite relationship
between cellular injury and functional disorganization of the metabolic system. One
could say that the factors that interfere with or excessively modify cellular
organization and cellular processes can be considered in a biological sense to injure or kill
the cell. Such interference could result from the interaction of cellular functional
entities with a multitude of agents, for example, absorption photones, collision with
elemental particles, interaction of its various molecules, etc.
As a result of these interactions, in terms of functional sub-units of the cell, the
following changes could take place: inactivation or alteration of enzymes;
modification of messenger and transport RNA; inactivation of genes; alteration of
metabolites, intermediates, and co-factors. Since various agents and molecules can interact
differently with cellular functional entities and molecules, a multitude of modes of
functional disturbances may develop, depending on the character of the agent
involved. It can also be considered from a chemical point of view that the molecular
alterations produced within the cell may be reversible or irreversible. Furthermore,
metabolic damage produced in the cell can be repaired by the ceil, or if this is not
possible, then the cell will die. However, at certain levels of injury the cell may
survive when it can compensate for the deficiencies itself, or when external agents are
applied which help cellular recovery. Consequently, we can generally define cellular
injury as a phenomenon in which normal functional processes of the ceil are altered
by interfering agents. Since various operational entities of the cell have specific
molecular composition and organization, the chemical interaction specificity determines
which of the operational units are altered by a particular agent. The specificity of the
agents may determine their character of functional injury. This can be a singular
injury in a single locus, or it can be a multiple injury in many locl. For example,
injury on a single gene level would be sufficient to kill the cell if this gene were
irreversibly inactivated. By contrast, when several of the enzymes are inactivated, we
have widespread functional damage, and it is not at all obvious how the cell would
be affected by such injury. As a matter of fact, cellular processes are so complex and
so interwoven that it is very difficult to perform a cleancut analysis of the system
unless the system is analyzed quantitatively. It is of interest that the cellular
modelsystem which has been developed can serve as useful purpose as a means of analyzing
the problem of cellular viability as well as an aid to studies on growth regulation
and growth in general.
In order to gain more insight into cellular disorganization, it is essential that a
systematic study be made on the effects which can be produced by interfering with the
activity of important functional entities in the cells. Furthermore, it would be
important not only to view the problem of cellular disorganization from a
phenomenological point of view but also to study the level and degree of disorganization.
B a s i c p r e m i s e s
a n d
p r o c e d u r e s
Quantitative studies on the ceil growth model reveal that the system is sensitive
to alterations at the level of any functional entity and that the change which occurs
is followed by adjustment at all levels of functional entities in the model-system.
Therefore, in order to understand the events that are taking place aflcer the
interaction at a particular site, it is essential that all changes occurring in other entities be
recorded simultaneously. Such a computer study is equivalent to the study of cell
physiology under conditions in which, simultaneously with the cellular injury,
physiological and biochemical measurements are made with all possible entities which
enter into the scheme. This is essential since the measurements of a single entity are not
sufficient to characterize the kinetic events in the cellular metabolism after a certain
type of injury. However, in many practical conditions in cellular biology, injury
occurs simultaneously at multiple sites. This is especially true in cases when a
heterogeneous agent is introduced into the system, for example, radiation, heat, etc. No
attempt is made to explore the multiple injury pattern, not that it would be too
complex to be analyzed, but rather it is too involved and needs a special programming
for that purpose. Here only a limited set of experimental data can be presented. A
more complete set of treatment of this subject will be presented elsewhere HEINMETS
I n a c t i v a t i o n
o f f u n c t i o n a l
t e m p l a t e s
Since genes represent only a small fraction of total cellular elements, it is evident
that non-genetic injury plays a much larger role than genetic injury. Due to limited
space we cannot analyze non-genetic injury at many sites, and therefore we limit
ourselves in this study only to template type injury.
In our previous studies, it was shown that the template Js a suitable site for
exercising growth control (HEINMETS1966). One can pose a question: What would happen
Fig. 6: Continuous repression of templates g and Np. The effect on: template N, messengerM
and ribosome B
to the functional system if regulatory compounds such as hormones or chemical agents
were applied on the templates in excessive amounts for long periods of time?
Furthermore, are the effects on the system only temporary and always reversible, or would
there possibly be permanent damage produced by the regulatory mechanisms when
they are operational at an extreme level? Template inactivation results when substrate
si (eqs. 24, 26, Table 2) interacts with N and Np. Then substrate s'i can restore template
activity (Table 2, eqs. 25, 27). In the normal system where there is limited growth for
the model-system, these rate constants have zero value. Only when it is desired to
control growth at the template level are these rate constants introduced. This
mechanism is equally applicable to an external agent(s) introduced into the system which
causes specific template inactivation or activation.
Figures 6 and 7 show an experiment where an external agent si is introduced into
the system (at time indicated by the arrow). This compound interacts with templates
N and Np and converts these into inactive state. It is evident that there is an
immediate reduction of template N followed by a slower reduction of other entities. The
system will become finally nonfunctional since all entities will disappear gradually.
The experiment presented in Figure 8 shows that a system which is growing
normally can be disorganized when an external agent, si, is introduced into the system.
Observation time on the computer is about 7 generation times. Agent si is introduced
about at 3/4 of generation time. The first effect is rapid loss of template N. This is
followed by reduction of template M synthesis. The rate of enzyme E synthesis is
reduced, while pool P~ continues to grow at a lower rate. Other entities which have
been recorded (not shown here) all reveal a reduction of concentration in varying
degrees. If the system is le~ indefinitely in such a condition, the model-system
becomes extinct since all entities finally cease to exist. It was of interest to study the
effect of an activating agents s'i when this is introduced into the system. Figure 8
shows the effect in conditions where si is removed and s'i is simultaneously introduced
into the system. It is evident that there is immediately a rapid increase of template N
concentration, while other entities start to grow subsequently. This experiment shows
that a decaying model-system may be restored into the normaI functional state when
a template activating compound is introduced into the system. However, other
experiments reveal that the functional system may be restored only in conditions where
activating agent is introduced before a "critical time limit", otherwise system cannot
recover. This experiment shows that system's survival is conditional, depending on the
duration of injury.
i n j u r y
Basically, genetic injury can be reversible or irreversible. In irreversible injury,
gene activity is partially or completely destroyed, and this effect is permanent.
Reversible injury can be corrected either by removal of the agent which interacts with the
gene or by the action of repair mechanisms. For example, if there is a complexing
process between an agent and the gene, then the [gene-agent~-complex can be
maintained as long as the agent is present in the system. H o w e v e r , when the agent is
removed, the complex dissociates and the gene is free to operate again. Another
aspect of genetic injury is t h a t it can be either partial or complete. In the latter case,
the gene is completely blo&ed and is not functionally operative. Partial injury means
Fig. 9: The effect of complete genetic block on gene GO level. At the time indicated by first
arrow, k~ is made zero. At time indicated by second arrow, the genetic blo& is removed.
Duration of genetic blo& is such that cell is capable of recovery. Pools Pi, Pa, and Pn are
that the gene is operational, but it does not function at the n o r m a l rate. This
condition can arise when the special configuration of the genetic structure suffers some
minor modifications. In partial genetic injury, the average genetic activity is reduced.
Furthermore, we assume that when the gene is blocked, it is not operational and does
not produce R N A . We do not analyze the case where the gene is producing a
defective, non-functional R N A . This can be done, but it requires some special programming
and model-system changes.
Genetic injury can be produced at any gene level presented in the model-system.
H o w e v e r , due to limited space we can treat the problem only at one gene group level.
Further details are available in another publication (HEINMETS 1966). H e r e we
consider genetic injury at gene G c level, whose function is to synthesize transport R N A C.
The rate constant k4 determines the activity of the gene. Here only complete gene
activation is considered. When a complete genetic blo& is introduced at 8/4 of
generation time, the components of the system start to decline and all the system becomes
extinct. However, when the genetic blo& is removed at a certain interval, system may
recover. Figure 9 shows experiments where various pools are recorded during 8
generation times. Other entities have been recorded, but only their behavior is discussed
here. The first event taking place when the gene is reactivated is the increase of
transport R N A C concentration. This is associated with the decrease of pool P~ and pool
Pa while pool Pi starts gradually to increase (Fig. 9). Practically at the same time an
increase of enzyme E and ribosome B occurs. In contrast, Mp and Ep both continue
to decrease. Both templates N and N~ were near the maximum concentration when
k4 was restored to the original value, and both started immediately to decline.
Template N declines initially very rapidly, but after a while reduction proceeds at a
slower rate. Messenger M shows initially practically no &ange, but subsequently there
is a decrease of M concentration. Aflcer passing a minimum, M gradually starts to
increase again. The recovery of the system takes place rather unevenly and various
functional entities which were reduced start to increase at different times. The first to
increase is enzyme Ep, then messengers M~ and M, followed by pool P~. Finally, at
the end of generation time, Np starts to increase while N reaches a constant level.
N finally also starts to increase, but this can be seen only when the observation time
is further extended on the computer. This experiment reveals that introduction of the
genetic bl0& really disorganized the system. However, removal of the blo& intiated
a sequence of events which permitted the system to recover. The recovery is not
uniform, however, and it wilt take numerous generations to stabilize the system. An
important conclusion which can be drawn from this simulation experiment is that a
cell may be capable of self-recovery from a prolonged genetic injury. In these
experiments duration of the genetic blo& was slightly longer than a generation time. This
raises the question of what the minimum duration of genetic blo& is which would
prohibit recovery of the system. In order to explore this, the experiment was
performed with the same model-system, but the growth rate was slightly reduced by
increasing ka0 and kit and by decreasing k18. This was necessary in order to avoid
computer overloading in transient phases of the experiments. In the absence of the
genetic block, this system starts to grow slowly and continues to do so indefinitely.
Rate constant k4 was reduced to zero at the same time as previously, but duration of
the zero period was markedly increased. Only pools are shown here in Figure 10. In
these conditions, ai%r removal of the genetic block, the system was not able to recover
and alI entities continued to decline. There is a small increase of ribosome B, but the
rise is very small compared with the previous experiments which recovered from the
injury. Also, transport R N A C has a very small increase in concentration, but it is
trivial and C does not continue to grow. Enzyme E has practically no increase, and
the concentration is maintained at a low level. Pool Pi remains constant for a while,
then starts to decline. The kinetic behavior of the system indicates that synthesis
initiated aflcer the removal of the genetic block is extremely small, and this synthesis
is not sufficient to make the system again operational. Some further experiments were
carried out on the computer to determine critical duration of the genetic block.
Experiments revealed that if the blocking time is slightly reduced, the cell is indeed capable
of self-recovery. Consequently, the experiment reported in Figure 10 represents rather
the borderline conditions where the system loses its ability to recover. It appears that
there is a critical time limit for a given system to recover from a particular type of
This simulation experiment represents a very important concept in cellular injury
analysis because it clearly reveals the significance of the time factor in the process of
recovery. However, if the genetic block is not complete, then different time
relationships appear and only systematic study can give useful information in this area. These
simulation experiments on the model-system suggest some considerations for cellular
systems. By an application of a temporary genetic block, the model-system is
terminated, and the question can be asked whether there is a requirement for permanent
genetic injury to kill the cells. It appears that perhaps a certain length of the blo&ing
time is sufficient during cellular growth to inactivate the cell. If this were the case,
then differential killing of different cell types could be possible simply on a time
basis, t f we assume that a long-lasting genetic blo& can kill off the cell, then the
question can be raised whether all cells in a population can be killed off with the
same critical bto& lasting a definite time. This, of course, seems possible if one assumes
that all cells behave equally. This assumption, of course, is contrary to practical
experience, since there are variations in various biological properties among individual
cells. This brings up the point of whether the properties of various functional entities
will effect the critical blo&ing time in the model-system and how much variation is
necessary. This problem, of course, would need a systematic study which is not done
here, but we shall present an example to demonstrate that the model-system can be
used to analyze such a problem. Furthermore, we shall show that stability variation
in the values of individual functional entities is the determining factor in determining
viability of the system.
k O -
Fig. 11: Same experiment as in Figure 10 except protein stability is increased. Rate constant
k~6 is reduced from .01702 to .0107. This increased enzyme E stability permits recovery of the
system. Pools Pi, Pn, and ribosome B are recorded
Figure 11 represents a similar experiment as in Figure 10 except that protein
stability has been increased by reducing rate constant k16. I t is evident that all pools
start to grow again afLer the removal of the genetic block. All other functional entities
(not shown here) also start to increase, and system becomes again functional.
OI1 the basis of the few experiments presented here and the extensive studies
published elsewhere (HEINMETS 1966), one can conclude that cellular recovery and
death are highly conditional phenomena. Many interdependent factors exert effective
control on such processes and more insight into basic operational mechanisms can be
gained by extensive research on the model-system basis. Model-system studies also
indicate that when severe damage occurs on the level of any entity, inactivation of
the system results (HEINMETS 1966). Obviously, the living cell is much better regulated
than a model-system. However, a basic concept which has been established on
modelsystem studies reveals that disorganization of the system can be produced on the level
of practically any functional entity, and extensive disorganization leads to decay of
the system. However, when interfering agent is removed, the cell may be capable of
initiating growth again, but the cellular fate is conditional. When such an experiment
is carried out in practice, the time factor, of course, will play an important role. If
conditions of injury are continued for a long time, cellular destruction will take place
because there is no compensatory synthesis. Any biological system will decay by the
inherent property that all functional entities are unstable, and in order to maintain
the system's maintenance, synthesis should occur. On the other hand, for a limited
amount of time, the cell can sustain a certain amount of decay and still be able to
recover when the condition for the growth is soon re-established. We should like to
make a few comments on cellular viability studies within a mass population on the
basis of experience which we have obtained by analyzing the model-system.
Modelsystem studies indicate that a viable system can sustain a certain amount of
disturbance. The resuiting disorganization depends upon how long the state of disturbance
of normal functional processes is maintained as well as on the stability of functional
entities of the system. It is a well known experimental observation that all cells do
not have exactly equal &aracteristics. For example, if a group of syn&ronized cells
are placed into growth medium, then within a few generations syn&rony is lost,
indicating that all cells are not equal in terms of metabolic and synthetic processes.
This is, of course, expected since the cell is a highly complex organizational and
functional system where statistical events in the terms of molecular interaction and
reactions represent the basic process. Consequently, the structure, as well as the
function of entities is probabilistic in nature. Furthermore, variations on the single entity
level are supplemented by sets of higher order interactions. Thus, systems develop
having a large number of probablistic states. Consequently, a population contains
species which have a wide variation in survival potential under normal conditions.
Subsequently when the cellular population is put under stress, one would expect that
there would be among individual cells a wide range of deviations from the average
behavior. If it is decided to destroy a cellular population completely, a process, for
example, whi& is essential for sterilization, then we may be able to kill off the
majority of population with a relatively small dose, but for the elimination of few
individuals whi& have high survival potential (extremely effective functional
organizations) a large dose may be required. We feel that on the basis of general concepts
emerged from computer studies on the individual celI model-systems, new theories
could be developed whi& could determine kinetics &aracteristic of population death.
REQUIREMENTS FOR COMPUTER PERFORMANCE AND LAYOUT
This is a mathematical problem. The model-system (Fig. 1) is represented by
nineteen simultaneous differential equations containing thirty rate constants. In order
that a functional system be established, the computer must have certain performance
characteristics as weli as a suitable physical layout. The organization of a functional
system on the computer is time-consuming and tedious. It also makes great demands
on the man who organizes the system, particularly in terms of memory, as well as
evaluation and pertinent intellectual reasoning during the operation. For a solution
to be obtained to such a problem, it is essential for the computer to possess certain
performance characteristics. We shall therefore outline a number of essential
requirements on the basis of our own personal experience:
(1) The computer must contain a sufficient number of operational elements
(amplifiers, integrators, multipliers, electronic switches, potentiometers, etc.). These must
perform with great dynamic accuracy. (2) The programming patchboard must be
removable so that the problem can be taken off the computer. This is imperative for
prolonged studies as well as for organizing the system. (3) All operational control
knobs and switches should be within easy hand-reach of the operator. (4) Computer
performance should have the following principal characteristics: (a) High speed, a
compressed time scale, a broad frequency band for accuracy and reproducibility. (b) It
must operate with high dynamic accuracy in real time, as well as in repetitive
performance. The latter is required for organizing the system by using an oscilloscope
for observations. I f the repetitive solution on the scope is not accurate, it is not
possible to organize the functional system. The real-time solution is used for recording
the data with an X - Y recorder. Switching from repetitive to real-time and vice versa
should be simple. (c) The computer should be able to maintain stable performance
when square-wave potentials are introduced into the system, and it must be able to
recover rapidly from overload conditions. (d) The noise level must be low. This is
extremely important when characteristics of the system are studied in conditions of
decay (for example, in studies of viability).
A D V A N C E D
M O D E L - S Y S T E M F O R CELL G R O W T H
It is desirable to extend model-system analysis to systems which contain more
basic growth regulation at genetic level. A more advanced model-system has been
developed (Fig. 12). Flow equations and a simultaneous set of differential equations
are represented in Tables 4 and 5. Such system can be analyzed at the time when more
advanced hybrid computers are available.
Factors "e" and " r " are variable modifiers which have maximum limit values.
- ' E 3 " '
28. Et -----~ Pi
1. Ep = k19 N~ [c Pa] - - k2o Ep - - k33 E~_~G+ k-33 [EpC] - - k34 Ep__BB+ k-34 [EBp] -
kl Ep.Pn + k'2 Gb [EpPn] + k"~ G,b [EpPn] + ks Gb [EpPn] + k3 Gc [EpPn] + k'4 G
[EpPn] + k7 Gp [E~Pn] + k8 Ge [EpPn] + kaGr [EpPn]
2. /~n = k31 kn E Pi - - kl EpPn + k'21 M + k23 Mp
3. [Ep'Pn] = ]el EpPn - - [EpPn] (k'2 Gb + k"2 Gb + k2 Gb + k3 Gc + k'4 G + k7 Gp +
k8 Ge -I- kg Gr
Gb -~ k 4 o e G b - - k 4 ~ G b
5. Gb = k41 r Gb -- k43 Gb
6. Gb -~ 1e42Gb +" k43 Gb - - k40 e Gb - - k41 r Gb
= [EpPn] (k2 Gb + k'2 Gb + k"2 Gb) - - k'll BM - - k14 - - BM p - - k24 B + k'16 N
[c Pa] + k19 Np [c Pa] - - k 3 4 EpB + k-34 [EpB]
-~ k3 Gc [EpPn] - - k15 c Pa + k'16 N [c Pal + k19 Np [c Pa] - - k25 C - - k33 EpC +
k _ 33 [EpC]
M = k'4 G [EpPn] - - k'll BM - - k'21 M + k'16 N [c Pa]
10. Mp = k7 Gp [EpPn] --k14 BMp + k19 Np [c Pa] - - k23 Mp
= k8 Ge [EpPn] - - k40 Gbe - - k44e - - klO er
= k9 Gr [EpPn] -- k41 Gbr -- k45 e -- klo, , e.. r
13. N = k'll B M - - k 3 8 N
+ k 3 9 s " i N * - - k ' 1 6 N [c Pa]
14. Np = k14 BMp - - k36 Np + k37 s'i Np - - k19 Np [c Pa]
16. ]'/~"=kas N - - ka9 s"i N*
17. /~a = ka k3"2PiEa - - k15 c Pa
18. [c Pa] = kl~ c P ~ - - k'16 N [c P~] - - k19 Np [c P~]
19. E = k't6 N [c Pa] --k'~6 E - - ka5EP? + k _ ~ [EPi]
20. / ~ i = k a o k t P e E = k a l k n E P i - - k a 2 k a E P i - - k 8 5 E P i + k - 3 5
k25 C + k'~6 E + k~9 Ep
[EPi] + k ~ 4 B +
21. [E;C] = k3a E p C - - k-a~ [EpC]
22. [E:~B] = k34E p B - k_ a4 [EpB]
23. [E-'P~] = k35 E P i - - k-35 [E Pi]
S U M M A R Y
1. A functional model-system has been helpful in elucidating the problem of cellular
growth kinetics. Computer studies with the model-system have made it possible for
us to obtain some information in regard to interrelationships between various
functional entities as well as on the effect of various parameters on the growth process.
2. These studies include the effects of the various entities on normal growth. The
phenomenon of cellular viability and the recovery of viability has been elucidated,
and it has been shown that the organization pattern plays a dominant role in a
3. While the simulation of the cellular system with the model has produced some
highly interesting results, the complexity of the problem, raises m a n y new issues.
In order to make a model-system analysis more penetrating for gaining insight into
biological processes, the model-system has to be expanded. I t should especially
contain more regulatory mechanisms.
4. In our experience, it was extremely difficult to obtain a solution on the computer
for the differential equations representing the model-system. Sometimes it was
thought to be impossible. Consequently, one has to introduce new- features into the
5. Since a functional system has been obtained, however, there is no apparent difficulty
to supplementing within certain limits a d d i t i o n a l features involving regulatory
mechanisms at various levels. This can be done with available computer facilities.
6. More expanded model-systems will also require new computer developments. H e r e
again, there is no a p p a r e n t limitation from the computer technology p o i n t of view.
The p r o b l e m is m a i n l y economic.
A C K N O W L E D G M E N T S
In these studies, the GPS Instrument Company (Newton, Mass., U.S.A.), General
Purpose Computer, Model GPS 10000, was used.
L I T E R A T U R E C I T E D
Discussion following the paper by' H~iNMETS
Hess: Which reaction triggers gene activity? How do you get the genetic level going?
HEINMETS: The model system for cell growth is analyzed during one growth cycle. Genetic
division is not considered here. All gene groups are active in the beginning of the growth
cycle. The problem of initiation of genetic activity by trigger mechanisms has been discussed
(H~IN~ETS, Analysis of normal and abnormal cetl growth. Plenum pr. New York,
BANSON: In the solution of such a complex system, we have found that many sets of
parameters lead to the same conclusion. How then do you decide among the various sets?
H~INM~TS: Model system is formulated by 19 time dependent differential equations, containing
33 rate constants. Here indeed exist many possible solutions. On the basis of general
experimental information, we have assigned certain values to rate constants and the analog computer
solution is obtained within the framework of such numerical data.
S~GITA: One of the difficulties in this work may be to get a suitable function generator. Why
are you using the ultra-high speed analogue computer?
HEINMETS: In this work we do not need a function generator because all entities in the model
system are derived by analog computer from the basic inter-relationship between the
functional entities. The computer is used as a mathematical tool to solve the set of differential
equations. It has to have high speed and accuracy in order to get solutions rapidly enough for
visual observation on oscilloscope (at least 20 solutions per second are required for such a
purpose). Visual observation of the complete solution on the scope is essential for obtaining a
functional model system.
KI~OG~R:Sie haben den Versuch unternommen, ausgehend yon Elementarvorg~ngen, die
Stoffumsetzungen im Organismus zu analysieren. Ida selbst versuche zun~chst die Ver~inderungen
im Makrobereich m~Jglichst exakt zu beschreiben und yon hier aus analytisch zu den
Elementarvorg~ngen zu kommen. Hier scheinen sich einfachere mathematische L6sungen zu ergeben. Diese
Bem~ihungen sind auch fiir die t~igticheArbeit des Biologen wertvoll.
HEss: I admire Dr. HEINMET'Swork, which I have followed for a long time; he has greatly
advanced the study of group systems. In our work we concentrated more on specific chemical
reactions. We started to construct a mathematical model of this type by writing down the
chemical equations representing each of the glycolytic steps including the enzyme substrate
interactions and product substrate interactions, In this system we write down about 120
equations and let the digital computer solve them.
KiErrt~: (1) I do agree that the target theory cannot account for many facts we have found in
radiation biology, but it is the only approach we have so far. Can your treatment give an
explanation for the survival curves commonly found? (2) You told us that recovery is
probably dependent on protein synthesis, but recovery processes a~er irradiation are not brought
about by protein synthesis. This seems to be inconsistent with your findings.
HEINMETS: Our analysis of cellular viability via the model system can provide an explanation
for the survival curves of cell populations. However, the problem is mathematically too
complex to be presented here. Model system analysis of cellular recovery al%r a reversible type
of injury revealed that protein synthesis is essential. However, from literature it is wetI known
that many injuries are irreversible (especially extensive genetic disorganization) and it is not
expected that protein synthesis is effective here. It was also pointed out during my presentation
that when cellular injury is extensive, the ability of the cell to synthesize some protein does not
restore the viability.
HEss: What is the time scale in your model; is the conservation law fulfilled? The digital
computer model, which I have developed with B. CHANCE and D. GARHNKEL,is based on
experimental data and well-known biochemical reactions, which are treated in form of common
mass action kinetics. Thus, it is possible to compute data of cell metabolism on the basis of
well-known facts. We do not need a complicated analog technique but can rety on appropriate
computer programs which handle the complex and otherwise rather unapproachable
HEINMETS: The basic time scale in our experiments is generation time. During that time all
functional entities of the model system have to double their value. It is considered that
subsequently cellular division takes place and all entities again acquire the unit value.
LOCKE~: Ich m~Schte gerne die Meinung yon Dr. HEINME'rS iiber den fi]r die
Computer-Anwendung so wichtigen Begriff der ,,Simulation" h6ren. Wenn wir auf der Basis gewisser
Kennmisse iiber ein System ein theoretisches Modell ftir dieses formulieren, Iegen wir bereits
lest, welche experimentell prtifbaren Resultate ein solches Modell liefern soll. Durch Vornahme
einer indirekten Messung, n~.mlich durch Untersuchung der Antworten des Modells auf
verschiedene inputs pr[ifen wir seine Brauchbarkeit. Kommt es hierbei zu einer nicht vertretbaren
Differenz zwischen den vom Modell vorausgesagten und den tats~chlichen experimentellen
Werten, mug die Komplexit~it des Modells erh/Sht werden, und zwar solange, his eine
signifikante Verminderung der Abweichungen zwischen vorhergesagten und beobachteten Werten
eintritt. Gibt es nun einen prinzipiellen Einwand gegen die Verwendung des Begriffes
,,Simulation" fiir dieses Verfahren?
HEmMrTS: There is no basic objection to use the word "simulation" in describing models.
However, it is essential that it should be made clear what kind of simulation procedures are
used and whether the method is adequate to simulate a complex system. For example, in many
occasions in the past, complex biological processes were simulated directly via electronic
circuitry. In many cases, such simulation is so crude that hardly any useful information can be
obtained. This is especially so when kinetics of the system are considered. A complex model
system, when represented via time-dependent differential equations, can yield informative data,
provided that adequate computing facilities are used. In my opinion there should always be
an interplay between experimental work and model system development.
BALAZS: (1) Welche Art genetischer Verletzungen wurde gew~ihlt und was verstehen Sie unter
der Behebung eines genetischen Blocks? (2) Bei atmungsgeschiidigten Strahlungsmutanten der
I-tefezellen ist eine Art Selbstregulation durch Rilckmutation zu beobachten. Wie w~ire es
mSglich, dies mathematisch zu deuten? (3) Es wurde beobachtet, dal] bei atmungsgesch~tdigten
Zellen der genetische Block dutch andere enzymatische Wege kompensiert werden kann,
wodurch geniigend Energie freigesetzt wird. Wie kSnnte man ein solches System mathematisch
HHNMEVS: (1) Genetic injuries were produced in the model system by converting an active
gene group into inactive state. This was accomplished on the analog computer by using special
electronic switches which were operated in synchrony with a computer solution. A~er a
desired time interval, the inactive gene was made again functional and the effect was observed
on the computer solution. Genetic inactivation was studied on ali gene groups presented in
the model system. (2) Radiation mutation processes can be simulated on the analog computer,
but a proper model system has to be established for such purpose. (3) Respiration damage and
its compensation via other enzymatic pathways presents an interesting problem. In principal,
such systems a r e h i g h i y suited for analog computer analysis, provided there is enough
information to develop a plausible and adequate model system.
HAWKINS: To set up an analog of a car suspension is difficult and requires a specialist. Once
the computer is set up, anyone can study the effect of, say, altering the stiffness of one spring.
Is it true in regard to your analysis that once the difficult part of setting up the computer is
done, another biologist or physiologist could use it to study, for example, enzyme inactivation?
HEINiVIETS:Once a model system has been developed and a solution obtained on a computer,
any competent biologist can carry out further experiments on the computer (perhaps with the
aid of a computer technician) and test out properties of the system.
MAKMASSE: I would like to return to Dr. BALAZS'second question. Would it be possible to
translate mathematically a genetic mutation, in Dr. HEI~,ZMETS'scheme, by a change in rate
HEINMETS:Yes, a rate constant change can represent genetic mutation.
ZERBST: Wie die Frage yon Herrn LOCKERzeigt, ergeben sich Schwlerigkeken mit dem Begriff
der Simulation. Entspricht sie einer homomorphen Wiedergabe eines Vorganges ohne
energetische und stoffliche Analogie? Vielleicht wtirde man besser Analogrechnung an Stelle yon
Simulation sagen. Bei den mathematischen Modellen besteht die Gefahr, dat~ wlr sle zu einfach
oder zu komplizlert ansetzen. Leider verfiigen wlr heute noch nicht fiber geniigend
Informationen hinsichtlich der L&ungsr~iume. Hier w~ire, neben der Zusammenarbeit mlt den
Physiologen und Biochemikern, eine Zusammenarbeit mit den Morphologen erforderllch.
HrII'~METS: It is quite true that the organization of a complex model system, which includes,
in addition to symbolic interactions, also geometry and energetics, is indeed difficult.
Collaboration between various specialists is required to develop and solve advanced model systems.
BA~ENDSEN , G. W. , 1961 . Damage to the reproductive capacity of human cells in tissue culture by ionizing radiations of different linear energy transfer . In: The initial effects of ionizing radiations on cells . Ed. by R. J. C. Harris . Acad. pr., New York, 183 - 194 .
DAVIS , B. D. & FEINGOLD , D. S. , 1962 . Antimicrobial agents: mechanism of action and use in metabolic studies . In: The bacteria. Ed. by I. C. Gunsalus & R. Y. Startler. Acad. pr., New York, 4 , 343 - 397 .
GRAY , L. H. , 1961 . Mechanisms involved in the initiation of radiobiological damage in aerobic and anaerobic systems . In: The initial effects of ionizing radiations on ceils . Ed. by R. J. C. Harris . Acad. pr., New York, 21 - 44 .
HAYN~: S , R. H. , 1964 . Molecular localization of radiation damage relevant to bacterial inactivation . In: Physical processes in radiation biology . Ed. by L. G. Augenstein , R. Mason & B. Rosenburg. Acad. pr. , New York, 51 - 72 .
HEINMETS , F. , 1954 . Paper presented at: Physiological concept of cellular injury and death . AAAS (Am. Ass. Advmt Sci.) Symposium , Berkeley, Calif. (Not publ.) - - 1960 . An analysis of the concept of cellular injury and death . Int. J. Radiat. Biol . 2 , 341 - 352 .
- - 1964a . Analog computer analysis of a model-system for the induced enzyme synthesis . J. theor. Biol . 6 , 60 - 75 .
- - 1964b . Elucidation of induction and repression mechanisms in enzyme synthesis by analysis of model systems with the analog computer . In: Electronic aspects of biochemistry. Ed. by B. Pullman . Acad. pr., New York, 415 - 479 .
- - 1966 . Analysis of normal and abnormal ceil growth. Model-system formulation and analog computer analysis . Plenum pr. , New York (in print).
LEA , D. E. , 1947 . Actions of radiations on living cells . Univ. pr. , Cambridge, 402 pp.
MARCOVICH , H. , 1961 . On the mechanism of lethal action Of x-rays and Escherichia coli K 12 . In: The initial effects of ionizing radiations on cells . Ed. by R. J. C. Harris . Acad. pr., New York, 173 - 182 .
PowEas , E. L. , 1962 . Considerations of survival curves and target theory . Physics Med. Biol . 7 , 3 - 28 .
RAIN , O. , 1945 . Injury and death of bacteria by chemical agents . Biodynamica 3 , 1 - 183 .
WEUB , R. B. , 1964 . Lethal effects of high and low temperatures on unicellular organisms . In: Physical processes in radiation biology . Ed. by L. G. Augenstein , R. Mason & B. Rosenburg. Acad. pr. , New York, 267 - 285 .
WOOD , T. H. , 1953 . Lethal effects of high and low temperatures on unicellular organisms . In: Advances in biological and medical physics. Ed . by J. H. Lawrence & J. G. Hamilton. Acad. pr. , New York, 4 , 119 - 163 .