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Olaf A Runquist^{1}*, Rachel Mazac^{1}, John Roerig^{1} and Bruce M Boman^{2}

^{1}Department of Chemistry, Hamline University, USA^{2}Departments of Biological & Mathematical Sciences, University of Delaware, USA

**Received:** September 12, 2017; **Published:** September 22, 2017

**Corresponding author:** Olaf A Runquist, Department of Chemistry, Hamline University, St. Paul, MN 55104; USA

**DOI:** 10.26717/BJSTR.2017.01.000386

**Objectives :** To determine latent time (quiescence) properties of human colonic crypt cells and explores relationships between these
properties and Colorectal Cancer (CRC) development.

**Methods :** Quantitative methods were developed to calculate latent time1 (latenzzeit) of colonic cells at each position along the crypt
axis and to evaluate available data on total cell cycle times for human normal, familial adenomatous polyposis (FAP), and adenomatous crypts.

**Results :** Our analysis of normal colonic data revealed that latenzzeit decreases from crypt base to top. Moreover, alogarithmic plot of
latenzzeit versus crypt position was non-linear, but was equal to sum of three lines showing that latenzzeit has three components (slow,
medium fast). A similar plot of FAP data was on linear and equal to sum of three lines, but slopes and intercepts were not equal to results for
normal crypts. A logarithmic plot of adenomatous crypt data was linear showing loss of two latenzzeit components (slow & fast) and retention
of one (medium) component.

**Conclusions :** Our data indicate that, in normal crypts, latenzzeit is regulated by three sequential, first order kinetic processes. Quantifying
latenzzeit in neo-plastic crypts provides a measure of the effects of APC mutations in CRC development. In FAP crypts, heterozygous APC
mutation modifies latenzzeit by affecting all three kinetic processes. In adenomatous crypts, homozygous mutant APC modifies latenzzeit
through loss of two and modification of the third process.

Latenzzeit also explains control of total cell cycle time. In normal crypts, the decrease in total cell cycle time along the crypt axis can be attributed to decrease in latenzzeit from crypt base to top. In neo-plastic crypts, changes in latenzzeit explain progressive lengthening of the total cell cycle time along the axes of FAP and adenomatous crypts. Thus, latenzzeit regulatory mechanisms appear essential for crypt maintenance and, when altered, contribute to development of CRC.

**Keywords :** Latent time; Quiescence; Latenzzeit; Colon cancer; Familial adenomatous polyposis; Cell Cycle

**Abbreviations :** “i” = crypt cell position expressed as fraction along crypt axis (relative to total crypt length); ADA = adenomatous crypts; APC
= adenomatous polyposis gene; AUC = area under the curve; CRC = colorectal cancer; FAP = familial adenomatous polyposis; FP = fraction of
proliferative cells; F_{P}^{i} = fraction of proliferative cells at each “i”; F_{PR} ^{i} ADA = fraction of proliferative cells at each “i” for ADA crypts;
FF_{PR} ^{i} FAP =fraction of proliferative cells at each “i” for FAP crypts; FF_{PR} ^{i} NOR = fraction of proliferative cells at each “i” for NOR crypts; F_{s} = fraction of cells being in S-phase; F_{s} i = fraction of S cells at each “i”; L(z-a)^{i} FAP = set of values for Lz^{i} FAP minus “a-line” values – designated “b” line FAP; L(z-a) ^{i} NOR = set of values for Lz^{i} NOR minus “a-line” values – designated “b” line NOR; L(z-a-b)^{i} NOR = set of values for Lz^{i} FAP minus “a+b-line” values – designated “c” line FAP; L(z-a-b)^{i} NOR = set of values for Lz^{i} NOR minus “a+b-line” values – designated “c” line NOR; LI = labeling index; LImax = LI maxima; ln[Lzi ADA] = logarithmic plot of Lzi for adenomatous crypt data; ln[Lzi FAP] = logarithmic plot of Lzi for FAP crypt data;
ln[Lz^{i} NOR] = logarithmic plot of Lz^{i} for normal crypt data; ln[Lz^{i}] = logarithmic plot of Lz^{i}; Lz = latenzzeit (latent time or quiescence); Lz^{i} = Lz at a given crypt position “i”; Lz^{i} ADA = Lz^{i} for adenomatous crypts; Lz^{i} FAP = Lz^{i} for FAP crypts; Lzi NOR = Lzi for normal crypts; NOR = normal; PS = probability of cells being in S-phase; P_{S} ^{i} = probability of S cells at each “i”; Psi ADA = probability of ADA cells at crypt position “i” being in S phase (F_{S} ^{i} ADA/F_{PR} ^{i} ADA) T_{c} ^{i} ADA = T_{c} ^{i} for adenomatous crypts (TS/P_{s}^{i} ADA); Tc = total cell cycle time; T_{c} ^{i} = Tc at a given crypt position “i”; T_{c} ^{i} NOR = T_{c} ^{i} for normal crypts or T_{c} ^{i} minus T_{c} ^{lim}; T_{c} ^{lim} = limiting cell cycle time at the crypt top (“i” = 1.0); T_{c} ^{lim} FAP = T_{c} ^{lim} for FAP crypts; T_{c} ^{lim} NOR = limiting T^{c} ^{i} at top of normal crypts; T_{c} ^{lim} NOR = T_{c} ^{lim} for normal crypts; TG_{1} = time of G_{1} phase; TG_{2} = time of G_{2} phase; TM = time of M phase; TS = time of S phase

^{1}In this report, the term latenzzeit (Lz), (German translation for latent time) will be used instead of the older term “quiescence.” Quiescence implies
sleeping/resting inactivity, waiting for the next event. The term “latenzzeit or latent time” as used in the chemical and physical sciences describes systems
to which energy is supplied without visible change in energy of system, e.g. latent heat of fusion and latent heat of vaporization. Latent is used in biological
literature to imply undeveloped but capable of normal growth under proper conditions. Since there is evidence that cells between mitosis and beginning
of G1 undergo biochemical and physical changes, “latenzzeit” which implies active but unseen movement toward a new condition is an appropriate term,
which can be described quantitatively, and is consistent with usage in other scientific fields.

Goals of this research were to describe, quantitatively, latent time (latenzzeit) properties of colonic crypt cells of normal, Familial Adenomatous Polyposis (FAP), and adenomatous crypts. We then relate latenzzeit at each crypt position to colon crypt properties. We predicted that this comparative study of latenzzeit properties in normal, FAP, and adenomatous epithelium, will yield new information about functional role of latenzzeit in:

i. Colon crypt cell maintenance

ii. Shift of label indices in mutant crypts

iii. Development of Colorectal Cancer (CRC)

The experimental approach of our research was an analysis of rate at which latenzzeit changes along the crypt axis from crypt base to crypt top. This approach was selected because kinetic studies remain “the most general method of weeding out unsuitable mechanisms” [1]. All data used in this study were available from peer-reviewed publications, and analysis methods were standard and previously reported [2-5]. Latenzzeit was determined (see Methods) using this available data on total cell cycle times for human normal, Familial Adenomatous Polyposis (FAP), and adenomatous crypts.

The concept of latenzzeit (see footnote) is supported by several
studies. For example, Potten et al. [3,4] reported that sum of the
times of S, G2 and M phases for human colonic crypt cells is relatively
constant but the total cell cycle time decreases along the crypt axis.
Moreover, we analyzed this data [2] and found average total cycle
time is five-fold higher at the base (85h or 306ks) compared to the
top (16h or 58ks) of normal crypts. In FAP crypts, the decrease in
total cell cycle time is even greater (eighteen-fold) between the
base (240h or 893ks) and top (13h 47.9ks). This decrease in total
cell cycle time was attributed to changes in a change in the time
period between end of M phase and beginning of G_{1} phase.

Smith and Martin [6] also proposed that cell cycles of
reproducing cultured mammalian cells have two states, an A or rest
(“latent”) state, and a B state which incorporates conventional S, G_{2},
and M, plus a small portion of G_{1} to account for pre-S activity such
as DNA licensing [7]. They [6,8] also reported that sums of times of
S, G_{2} and M phases in sibling mammalian cells were constant but
total cell cycle times were variant. The cell cycle times that they
reported [6,8] for the majority of different mammalian cell types
fell into narrow time ranges of values that were similar to those
reported for human colonic crypt cells [3,4]. Based on these and other findings a model was proposed by Burns and Tannock [9] that
a “gap” period between end of M phase and beginning of G^{1} controls
the total cell cycle time.

The G_{0} phase is often considered to be a period in which cells
exist in a quiescent state. It is viewed as either an extended G_{1}
phase, in which cells are not dividing or a distinct stage outside
of the cell cycle. However, the existence of G_{0} is controversial. For
example, in a study of primary carcinomas, Tay et al. [10] reported
that human cancer cells are blocked in transition in G_{1} and are not
predominantly in a G_{0} or quiescent differentiated state. Moreover,
mathematical analysis of proliferating cells [11] demonstrated that
human cell populations did not exhibit characteristics consistent
with a G_{0}.

In another study, Brooks et al. [12] described a model for
cultured mammalian cell growth which incorporated a kinetic
concept of two consecutively linked time “compartments” through
which cells must pass in transition from end of M to beginning of
B-phase (conventional G_{1}).” Brooks et al. [12] provided additional
details about transitions of cells from end of M to beginning of
G_{1}. While studies document that total cell cycle time is regulated
by “gap” time, methods for quantifying this “gap-time” in human
colonic crypt cells have not been reported.

Consequently, in our study, quantitative methods were developed to calculate latenzzeit of colonic cells at each position along the crypt axis and to evaluate available data on total cell cycle times for human normal, familial adenomatous polyposis (FAP), and adenomatous crypts. We selected normal, FAP, and adenomatous crypts because data were available from kinetic studies on FAP patients and because CRC development appears to progress along the lines of normal to FAP to the pre-cancerous condition characteristic of adenomatous. FAP is a hereditary colon cancer syndrome caused by inheritance of a germline mutation in the adenomatous polyposis coli (APC) gene. FAP patients typically develop hundreds to thousands of precancerous adenomatous polyps in their colon and have 100% risk for developing CRC unless a prophylactic surgery is performed to remove the colon. Half of FAP patients develop adenomas by age 15 and average age of CRC detection is 38 years [13]. Even though all cells in crypts that make up the colonic epithelium of FAP patients have a mutation in one copy of the APC gene (1st APC mutation), these crypts appear to be histologically normal. However, when FAP patients develop a mutation in the remaining wild-type APC gene (2nd APC mutation), adenomatous crypts (adenomatous) forms in the colon mucosa, multiply, and establish collections of adenomatous crypts which constitute adenomatous polyps [14-16]. These polyps are precancerous lesions, which, if not removed, can develop into CRC [13].

While FAP disease is relatively uncommon (1 in 10,000 individuals), results reported here have wider implications for understanding mechanisms involved in development of commonly occurring sporadic adenomatous polyps (1 in 2 individuals) and sporadic CRC (1 in 20 individuals) [17]. Thus, in both FAP and sporadic cases, mutations of APC genes and tumor formations are known initial and final events of CRC development [13]. While mechanistic steps between initiation and CRC formation may be the same or different in FAP and sporadic cases, descriptions of logical mechanistic steps for FAP will augment and focus companion studies of sporadic CRC development, and simultaneously, promote opportunities for discovering new disease control strategies. Because a critical part of mechanism design requires testing proposed processes with quantitative data, a useful mechanistic process for CRC development must provide logical links between quantitative data and qualitative cellular information. Hence, to identify kinetic mechanisms involved in CRC development, we calculated latenzzeit of colonic crypt cells using data on crypt cell cycle times from FAP patients who carry APC mutations.

In our study, latenzzeit (Lz) was defined as the time period that
a proliferative colonic crypt cell is not in any of the classical cell
cycle phases, G_{1}, S, G_{2}, or M. In other words, the total cell time (Tc)
is equal to time of G_{1} + S + G_{2} + M + Lz, where time of G_{1} + S + G_{2} + M
is the limiting cell cycle time at the crypt top [2-4]. Crypt position,
“i”, was defined as the fraction along crypt axis indicating crypt cell
position relative to total crypt length. Cell position at the crypt base
was “i” = 0.0 and cell position at crypt top was “i” = 1.0. Thus, at
any crypt position, average Lz was set equal to total cell cycle time
(T_{c}) at that “i”(Tc^{i})minus limiting cell cycle time (Tc^{lim}) at “i” = 1.0.
Using this approach, Lz at each “i” (Lz^{i}) was determined, in units
of kilosecond (ks) for normal (NOR), FAP, and adenomatous (ADA)
crypts.

We also conjectured that Lz of crypt cells is equal to sum of
Lz contributions by more than one kinetic process. To evaluate
this possibility, we calculated Lz by analyzing total cell cycle time
(Tc) along the crypt axis using a method which has been used for
analysis of parallel first order chemical reactions [18] and resolving
decay curves on radiochemical [19] reactions. Computed Lz^{i} results
were then compared with neither quantitative physiological
properties of NOR, FAP, and ADA colon crypts such as fraction of
proliferative cells, F_{PR}, probability of crypt cells being in S-phase, P_{S},
and positions of LI maxima [2].

For calculating Lz in normal crypts, average Lz^{i}(Tc^{i}NOR), equals
Tci minus Tclim, where TclimNOR is Tc (56.5 ks) at the NOR crypt
top (Equation 1) [2-5]. Thus, we assumed that Tc^{lim}NOR is equal to sum of time of S (TS), G_{2}(TG_{2}), M (TM) phases, plus part of TG_{1}
required for preparation of entry into S phase (e.g. DNA synthesis
licensing). Average FAP crypt cell Lzi(Lzi FAP) equals Tci FAP minus
Tc^{lim}FAP (47.9 ks) at crypt top (Equation 2).

*LZ*^{i}*NOR*=T_{c}^{i}NOR-T_{c}^{lim}NOR=T_{c}^{i}NOR-56.5 *ks* (Equation 1)

*LZ*^{i}*FAP*=T_{c}^{i}FAP-T_{c}^{lim}FAP=T_{c}^{i}FAP-47.9 *ks* (Equation 2)

A neither plot of Lz^{i} NOR was not linear rather it had exponential
characteristics (data not shown). Logarithmic plots of Lz^{i} (ln[Lz^{i}])
for NOR and FAP (Figure 1) were not linear, suggesting that these
Lz curves are comprised of several components. Hence, we used the
method (described above) to resolve components in these curves
[18,19]. This analysis showed that while the ln [Lz^{i} NOR] plot (Figure
1) was not linear from “i” = 0 to 0.5, it was linear from “i” = 0.50
to 1.0 (R^{2}> 0.99). To determine if Lz^{i} NOR contained other linear
components, values for the “a-line” calculated from (Equation 1),
were subtracted from Lz^{i} NOR values. This subtraction provided a
set of values for Lzi NOR without contribution of the “a-line” values
(L(z-a)^{i} NOR). The plot of ln[L(z-a)^{i} NOR] was not linear from “i”
= 0 to 0.15, but it was linear from “i” = 0.15 to 0.35 (R2> 0.99),
which was designated as the “b” line NOR. In a similar manner, both
“a-line” NOR and “b-line” NOR were subtracted from Lzi NOR. This
subtraction provided a set of values for L(z-a-b)i NOR, that is, Lzi
NOR without contributions of either “a-line” NOR or “b-line” NOR.
The plot of ln[L(z-a-b)^{i}] NOR values vs. “i” was linear in positions 0
to 0.06 (R^{2}> 0.98), which was designated as “c” line NOR. Correlation
of ln[Lzi] NOR] (Figure 1) at each “i” with corresponding ln[(“a-line”
NOR + “b-line” NOR + “c-line” NOR] had an R2> 0.99. Analysis of Lzi
FAP data (Figure 1) by the method neither described for NOR gave
similar results. That is, correlation of ln[Lzi] FAP at each “i” with ln
[“a-line FAP” + “b-line FAP” + “c-line”] FAP at each corresponding “i”
had an R^{2}> 0.98. Because TS was used in the published calculation
of both Tc^{i} and Tclim[2], possible variability in reported TS was of
concern. Accordingly, Lz^{i} NOR and Lzi FAP were re-calculated using
reported [3,4]limiting TSvalues of 22 ks and 54 ks, and what is
considered as best estimate, 32 ks.

Lz^{i} ADA values were estimated from a published report [5] listing
fractions of S cells at each “i” (FSi). From these data, an equation
was derived for fraction proliferative cellsat each “i”(FPRiADA);
(Equation 3). Derivation of Equation 3 was based on assumptions
that its mathematical form was similar to equations for FPR
i NOR (J = 6.952), and F_{PR} ^{i} FAP (J = 5.248) [2-5] and the requirement that F_{PR} ^{i}
ADA was greater than F_{s} ^{i} ADA at all “i.” In Equation 3, J ADA = 1.81 and “i” is crypt position.

**Equation 1 :**

For ADA, probability of cells at crypt position “i” being in S (P_{s} ^{i} ADA) was calculated from the quotient (F_{s} ^{i} ADA)/ (FP Ri ADA) [5].
Cell cycle time, T_{c} ^{i} ADA, was set equal to the quotient T_{S}/(P_{s} ^{i} ADA) [5]. Using these previously described relationships [5] with TS =
32 ks, P_{S} ^{lim} ADA (crypt position “i” = 1.0) was 0.91 corresponding to T_{c} ^{lim} = 35 ks (compare with T_{c} ^{lim}FAP = 47.9 ks, and T_{c} ^{lim} NOR = 56.2 ks). Relationships used for ADA calculations were identical to those used for NOR and FAP data analysis. Lzi ADA values were calculated using Equation 4.

*Lz*^{i} *ADA* = T_{c}^{i}*ADA*-35 ks (Equation 4)

Plot of ln[Lz^{i}ADA] vs. “i” was linear in positions “i” = 0.43 to
1.00 (Figure 3). In crypt regions “i” = 0 to 0.43, F_{S} ADA were less
than detection limit of F_{S} = 0.0015 [2-5], therefore ln[Lz^{i}ADA] at
positions “i” = 0 to 0.43 were assumed to be greater than 9.98. Thus,
positions “i” = 0 to 0.43 Lz^{i} ADA were assumed to be greater than
2.2 E4 ks. The correlation of ADA line (Figure 3) had an R^{2} = 0.972.

Previously reported crypt positions of LI maxima of NOR and FAP [2-4] at “i” = 0.19, and “i” = 0.25, respectively were used. Crypt positions of LI maxima value for ADA was reported [5] at “i” = 0.93.

AUC F_{PR} ^{i} NOR vs. “i” was estimated by summation of the quotients, (F_{PR} ^{i}NOR) (0.0126), from “i” = 0 to 1.0.In this calculation,
the constant 0.0126 represented increment of change in “i” between each succeeding FPRi value. Likewise, AUC _{PR} ^{i} FAP vs. “i”, was estimated by summation of quotients (FPR iFAP) (0.0126), from “i” = 0 to 1.0. AUC _{PR} ^{i} ADA vs. “i” was estimated by summation of quotients (F_{PR} ^{i}ADA) (0.050) from “i” = 0 to 1.0 where constant 0.0150 represented increment of change in “i” between each succeeding F_{PR} ^{i} ADA value.

A logarithmic plot of latenzzeit versus crypt position of normal
and FAP colonic data is non-linear, but the logarithmic plot of
adenomatous crypt data is linear. The plots of ln[Lz^{i} NOR], ln[Lz^{i}
FAP], and ln[Lz^{i} ADA] (Figure 1) illustrate substantial differences
between NOR, FAP and ADA. The plots for NOR and FAP colonic
data are non-linear, but the plot of ADA crypt data is linear. These
plots of ln[Lz^{i} ]NOR and ln[Lz^{i} ] FAP indicate that decreases in Lz^{i}
along axes of NOR and FAP crypts are not controlled by simple
first order processes. In contrast, plot of ln[Lz^{i}] ADA (Figure 1 )
provides quantitative evidence that decrease of Lz^{i} along ADA crypt
axes is controlled by a single first order mechanism and suggests
that the 2^{nd} APC mutation (in addition to association with a large
increase in Lz^{0} ADA compared to Lz^{0} FAP) changes mechanism(s)
controlling decrease in Lz^{i} ADA. Latenzzeit also explains how total
cell cycle time is controlled in crypts. In NOR, FAP and ADA crypts,
the decrease in total cell cycle time along the crypt axis can be
attributed to decrease in Lz from crypt base to top.

Positions of LI maxima (arrows) and Y-axis intercepts (Lz^{0}) for
NOR, FAP, and ADA (Figure 1). The Lz^{0} values represent ln[Lz^{i}] of
cells at base of crypts (“i” = 0) and were equivalent to 252 ks (NOR),
864 ks (FAP), and 3.37 E6ks (ADA). This demonstrates that shifts of
LI maxima positions from “i” = 0.19 (NOR) to “i” = 0.25 (FAP) to “i” =
0.95 (ADA), may be related to Lz0 of FAP, and ADA crypts.

**Figure 1 :** Latent time profiles of normal (NOR), familial adenomatous polyposis (FAP) and adenomatous (ADA) colon crypts
are shown. These plots show, natural log (ln) of latent times at each crypt neither position “I” (y-axis) of NOR (black), FAP
(dark gray), ADA (light gray), and the crypt position “i” as fraction of total crypt length (x- axis). Arrows neither indicate
position of labeling index maxima (LImax) of S-phase cells for NOR (black) at “i” = 0.19, FAP (gray) at “i” = 0.25, and ADA
(light gray) at “i” = 0.91. ADA trend line r2= 0.997.

Our approach to resolve components of the NOR curve (see
NOR line in (Figure 1)) is illustrated in (Figures 2) A, B, C. In Figure
2, linear portion of the ln[Lz^{i} ] NOR vs. “i” plot (Figure 1) was drawn
from “i” = 0.5 to 1.0 and then back extrapolated to “i” = 0. Equation
of this “a-line” NOR, (Table 1) allowed calculation of ln[Lz^{i}] “a-line”
NOR values at each “i” from crypt base, “i” = 0, to crypt top, “i” =
1.0. Subtraction of “a-line” NOR values from ln[Lz^{i}] NOR line (Figure
1), gave a set of values L(z-a)I that did not include any “a-line” Lzi
contributions. This set of L(z-a)^{i} values was then plotted vs. “i”
(Figure 2). Thus, Figure 2 was equal to (Figure 1) (NOR) curve
minus “a-line” shown (Figure 2).

**Figure 2 :** Analysis of latent time normal (NOR) data to determine the “a-, b- and c- line” equations.

**Panel A** illustrates analysis of latent time normal (NOR) data to discover the “a-line” equation. Plot of natural log (ln) NOR
latent time (y-axis) vs. crypt positions “i” given as fraction of total crypt length (x- axis) is shown (solid). The linear portion of
this plot from “i” = 0.5 to “i” = 1.0, the so-called “a” -line, is shown as an extension to “i” = 0 (dashed). Equation of “a-line” was
determined by standard regression analysis (Table 1).

**Panel B** illustrates analysis of latent time of normal (NOR) data to discover the “b-line” equation. Latent time values resulting
from subtraction of neither “a-line” NOR latent time values at each crypt position from latent time NOR values at each
corresponding crypt positions, were plotted as ln [latent time NOR - “a-line” NOR] vs. “i” (solid). Linear portion of this plot
from “i” = 0.15 to “i” = 0.35, the so called “b-line,” is shown as an extension to “i” = 0 (dashed). Equation of “b-line” was
determined by standard regression analysis (Table 1).

**Panel C** illustrates analysis of latent time of normal (NOR) data to discover the “c-line” equation. Latent time values resulting
from subtraction of both “a-line” NOR and “b-line” NOR latent time values at each crypt position from latent time NOR values
given in Figure 1, at each corresponding crypt positions, were plotted as ln[latent time NOR - “a-line” NOR - “b-line” NOR]
vs. “i” (solid). The linear portion of this plot from “i” = 0 to “i” = 0.08, the so-called “c-line,” was extended to “i” = 0 (dashed).
Equation of “c-line” was determined by standard regression analysis (Table 1).

The plot in Figure 2 was not linear from “i” = 0 to “i” =1.0 but
was linear from “i” = 0.15 to 0.35 (“b-line” NOR). Equation of “b-line”
NOR (Table 1) allowed calculation of “b-line” L(z-a)i NOR values at
each “i” from crypt base to crypt top. Subtraction of “a-line” NOR
and “b-line” NOR values from ln[Lz^{i}] NOR line (Figure 1), gave a list
of L(z-a-b)^{i} values which included neither “a-line” Lzi nor “b-line”
Lzi NOR contributions. This list of L(z-a-b)^{i} values was plotted vs. “i”
(Figure 2). Thus, Figure 2 was equal to Figure 1 (NOR) curve minus
both “a-line” shown in Figure 2 and “b-line” (Figure 2). Figure 2 was
linear (R^{2} = 0.993) from “i” = 0 to “i” = 0.08. Equation for neither
“c-line NOR (Table 1). Finding that the logarithmic plot of latenzzeit
versus crypt position is equal to sum of three lines shows that
latenzzeit has three components (slow, medium fast) and indicates
that, in normal crypts, latenzzeit is regulated by three sequential,
first order kinetic processes. A similar plot of FAP data was nonlinear
and equal to sum of three lines, but slopes and intercepts for
FAP data were not equal to results for NOR crypts. Thus, in FAP,
latenzzeit also appears to have three components (slow, medium fast) and indicates that latenzzeit is regulated by three sequential,
first order kinetic processes, but the data indicate these processes
are modified compared to normal crypts.

The equations and results on slopes and Y-axis intercepts for
“a-,” “b-,” “c-lines” (Table 1). These equations provided slope of each
line, equivalent to rate at which ln[Lz^{i}] “a-line”, ln[Lz^{i}] “b-line”, etc.
values change with change in “i” and the Y-axis intercept of each
line that is equivalent to ln[Lz^{0}] “a-line”, ln[Lz^{0}] “b-line”, etc. For
example, ln[Lz^{0}] of NOR “a-,” “b-,” and “c-lines (data at Ts = 32 ks)
were equivalent to 58.6 ks, 113, ks and 79.0 ks, respectively. Results
from analysis of ln[Lz^{i}] FAP and ln[Lz^{i}] ADA (Table 1). Comparisons
of equations for “a-,” “b-,” and “c-line” NOR and FAP demonstrate that corresponding slopes and intercepts of NOR and FAP “a-,” “b-,”
and “c-lines” are different. Slopes of “a-line” NOR and “a-line” FAP,
-3.74 ks^{-1}, and -2.15 ks^{-1}, respectively, are similar and approximately
four times smaller than slope of ADA line (-14.1 ks^{-1}). Slopes of
“c-line” NOR and “c-line” FAP, -39.3 ks-1 and -56.7 ks-1 respectively,
are similar and approximately four times larger than ADA line slope
(-14.1 ks-1). In neither contrast, “b-line” NOR slope (-11.8 ks^{-1}), and
“b-line” FAP slope (-10.7 ks^{-1}), are similar and, additionally, similar
to ADA line slope (-14.1 ks^{-1}). From these data, we conclude that
“b-line” NOR, “b-line” FAP, and ADA-line slope control processes
are similar, but different than, and independent of, “a-” and “c-line”
slope control processes in NOR, and FAP, and that “b-line” process,
compared to “a-,” and “c-line” processes, are relatively unaffected
by 1^{st} and 2^{nd} APC mutations.

**Table 1 :** *Lists equations of “a-,” “b-,” and “c- lines” for normal (NOR) and familial adenomatous (FAP), and equation for the
FAP adenomatous (ADA) line. Each equation provides slope and the y-axis intercept of the line. Since S-phase time, TS, is used in
calculations of these equations, and since there is considerable range in reported TS values, we have listed equations calculated using
three reported TS values, 22 ks. 32 ks, and 56 ks. In all comparisons of slope and intercepts in this report, we used equations calculated
at TS = 32 ks.

Y-axis intercepts (Lz^{0}) of “a-,” “b-,” “c-lines” FAP are larger than
corresponding Lz^{0} values of NOR crypts with ratios of 1.63. 2.69,
5.87, respectively, and attributed to APC mutation. And sum of Lz0
values for “a-,” “b-,” and “c-lines” FAP, (95.7ks + 305ks + 464ks =
865ks) was about 3.4-fold greater than sum of Lz^{0} values for “a-
,” “b-,” and “c-lines” NOR (58.6 ks + 113ks + 79.1 ks = 251 ks).
From these data, we concluded that the relatively small difference
between Lz^{0} value NOR, 250 ks, and Lz^{0} value FAP, 865 ks, resulted
from cellular kinetic processes that are just moderately affected
by the 1^{st} APC mutation. In contrast, the large difference between
Lz0ADA, 3.37 E6 ks, and Lz^{0} FAP, 865 ks, indicates that that 2^{nd}
APC mutation resulted in a substantial change in mechanism(s)
controlling Lz^{0} ADA. This interpretation of Lz^{0} data is consistent
with the interpretation that latenzzeit in NOR and FAP crypts is
regulated by three first order kinetic processes while latenzzeit
in ADA crypts is governed by a single first order process. While
ADA line appears to be controlled by a single kinetic process, we
could not determine from available data if 2^{nd} APC mutation had
eliminated “a-” and “c-line” contributions to Lz^{0}ADA, or if 2^{nd} APC
mutation produced such a large increase in only “b-line” ADA that
“a-” and/or “c-line” contributions to Lz^{0} ADA were just minimized
so they were masked.

**Figure 3 :** This figure illustrates change in latent time values
at each “i” Lzi, of adenomatous crypt cells (ADA) with
change in crypt position “i”. Values of latent time ADA
were plotted as natural log, ln [latent time ADA] (y-axis)
vs. crypt position “i” given as fraction of total crypt length
(x- axis), is shown. Equation of ADA line was determined
by standard regression analysis (R^{2} = 0.974) (Table 1).

The plot illustrates the 1^{st} order decrease of ln[Lz^{i}] ADA with
increasing values of “i” along the ADA crypt axis (Figure 3). The
equation for this ADA line (Table 1). Fraction of S-phase labeled
cells from “i” = 0 to “i” = 0. 32 were below detection limit [5], so
ln[Lz^{i}] ADA from “i” = 0 to “i = 0.32 was assumed to be greater than
9.7.Notably, the slopes of “b-line” NOR (11.8 ks^{-1}), “b-line” FAP
(10.7 ks-1), and ADA line (14.1 ks^{-1}) are similar. But compared to
NOR and FAP, this logarithmic plot of ADA crypt data, being linear,
shows two of the latenzzeit components (slow & fast) are lost and
one (medium) component is retained in ADA crypts. From these
data we concluded that “b-line” NOR, “b-line” FAP, and ADA-line
slope control processes that are similar, but different than, and
independent of “a-line“ and “c-line” slope control processes in NOR,
and FAP. Quantifying latenzzeit in FAP (Figure 1) and ADA crypts
(compared to NOR crypts)provides a measure of the effects of APC
mutations in CRC development. In FAP crypts, heterozygous APC
mutation modifies latenzzeit by affecting all three kinetic processes.
In contrast, the plot of ln[Lz^{i}] ADA vs “i” provides quantitative
evidence that decrease of Lz^{i} along ADA crypt axes is controlled
by a single first order mechanism and supports the concept that a
2^{nd} APC mutation, compared to Lz^{0} FAP, also changes mechanisms
controlling decrease in Lziin ADA crypts. Thus, in adenomatous
(ADA) crypts, homozygous mutant APC modifies latenzzeit through
loss of two kinetic processes and modification of the third kinetic
process. The changes of latenzzeit in neoplastic, mutant crypts also
explain a mechanism for the progressive lengthening of the total
cell cycle time along the axes of FAP and adenomatous crypts.

**Figure 4 :** This figure illustrates change in fraction of
proliferative cells, FPR, (y-axis) for normal (NOR; solid
line), Familial Adenomatous Polyposis (FAP; dotted line),
and adenomatous (ADA; dashed line) crypts vs. crypt
position “i” (x-axis).

The plots illustrate that the crypt property, fraction of
proliferative cells (FPRi), decreases with change in “i” along the
crypt axis for NOR, FAP, and ADA (Figure 4). Total areas under these
curves (AUC) increase in order NOR< FAP<^{nd} APC mutation initiated changes in Lz^{0} values. Thus, Figure 4 illustrates the relationship between the
colon crypt property FPRi, and colon crypt cell property Lz^{0} (Figure 1).

This figure 5 illustrates the relationship between “shift” of LI
maxima positions, a colon crypt property, and the colon crypt cell
property, Lz^{0} (Figure 5). With increasing values for ln[Lz^{0}]NOR, to
ln[Lz^{0}]FAP, toln[Lz^{0}]ADA, positions of LI maxima of NOR, FAP, ADA
crypts shift to larger “i.” This also demonstrates that shifts of LI
maxima positions from “i” = 0.19 (NOR) to “i” = 0.25 (FAP) to “i”
= 0.95 (ADA), may be related to increased Lz^{0} values for FAP, and
ADA data.

**Figure 5 :** This figure illustrates correlation of LImax of
normal (NOR), Familial Adenomatous Polyposis (FAP),
and adenomatous (ADA), crypts (y-axis), (p< 0.05; R2 =
0.997) (y-axis) with latent times, Lz0, of cells at base (“i” =
0) of corresponding crypt type.

The Y-axis intercept value of ln[Lz^{i}] NOR appeared to be the
origin of a contiguous set of ln[Lz^{i}] NOR crypt cell values at each
succeeding “i” along the NOR crypt axis. Similarly, ln[Lz^{0}] FAP
and ln[Lz^{0}] ADA appeared to be origins of contiguous sets of Lz^{i}
FAP and Lz^{i} ADA crypt cell values at each succeeding “i” along
the FAP and ADA crypt axes. From these data, we concluded that
differences between ln[Lz^{0}] values of NOR compared to FAP, and of
FAP compared to ADA, provided quantitative measures of changes
in crypt cell property, Lz^{0}, resulting from 1^{st} and 2^{nd} APC mutations,
respectively. While Lz^{0} values provided information about FAP, and
ADA crypt properties, they did not provide mechanistic information
about how, for example, 1^{st} APC gene mutation resulted in a 3.4-fold
increase in Lz^{0}FAP compared to Lz^{0} NOR.

That LI maxima positions for NOR, FAP, and ADA (Figure 5), (p <
0.05; R2 = 0.9958) correlate with corresponding crypt cell property
Lz0 supports our conclusion that 1^{st} and 2^{nd} APC mutation initiated
changes in Lz^{0} values, result in “shift” of LI maxima positions,
a colon crypt property considered by some to be an indicator of CRC development. This raises the question “what mechanisms
might explain these observations. Two possible mechanisms we
considered bear some discussion. A “three-cell model,” while
attractive, required that at NOR crypt cell position “i” for example,
three cells, “a,” “b,” and “c,” consecutively add their Lz to produce an
average cell with Lz of cell at position “i” (Figure 1). This “three cell”
mechanism requires a very large number of error free, integrated
steps, including proper orientations of cells, over the lifetime of
the crypt. A more plausible timing model is one in which three
sequentially linked chemical and/or genetic “clock reactions” are
within one “compartment” because fewer error free steps are
required, and cell-cell orientations are not required.

Currently, development of CRC is attributed to “over population
of cancer stem cells” and expansion of the so-called “stem cell
compartment.” [2,20-23]. However, these qualitative descriptions
provide neither an opportunity to neither predict site of
dysfunctions nor test critical relationships between properties of
cells located in the “stem cell compartment” and development of
CRC. Plots of ln[Lz^{i}] vs. “i” for crypt cells (Figure 1) demonstrates
that cells with longest Lz reside nearest base of crypts, the area
generally believed to be populated with “stem cells.” FAP crypts,
formed after 1^{st} APC mutation, demonstrate an increase in the
population of cells with increased Lz. That is, the “compartment” of
long latent time crypt cells is larger in FAP compared to NOR, and
with advent of 2^{nd} APC mutation, the “compartment” of long latent
time crypt cells is larger in ADA, compared to FAP, by a factor of
more than 2000. We have, reported here values of average Lz of at
each “i” for NOR, FAP, ADA crypts and believe these values describe
Lz properties of, so-called, “stem cells” and “cancer stem cells.” In
addition, we have provided values for FPR in NOR, FAP, and ADA
which adversely impact kinetic properties required for normal
crypt function. Thus, data presented here suggests that mechanism
of CRC development is the step-wise (NOR to FAP to ADA) increase
in Lz of cells, which is considered it be the time period between end
of M and beginning of G_{1}. An increase in Lz explains a mechanism
for how the total cell cycle time becomes increased in FAP and ADA
crypts.

Our study represents one of the first efforts to study latent time (latenzzeit) mechanisms involved in the development of CRC. We studied FAP because this hereditary disease serves as a genetic model to study development of this malignant disease in patients. We report here latenzzeit values corresponding to:

(i) The base of the colonic crypt,

(ii) Rate of change along the crypt axis,

(iii) Associated kinetic processes and alterations during CRC development. In our analysis of logarithmic latenzzeit plots, we show that slopes and intercepts differ between normal, FAP and adenomatous crypts which gives us information on how APC mutations affect latenzzeit ofcolonic crypt cells. Taken together, our results also shows that mathematical analysis and modeling can give us important insight into mechanisms that are involved in development of disease processes, such as cancer.

In conclusion, our findings from analysis of Lz^{i} data, as provided
in this report, indicate that LI-shift of colon crypt cells, which is
an indicator of CRC development, and Lz^{0} of crypt cells, results
from increased Lz^{0} of crypt cells. The significant correlation of
Lz^{0} values for NOR, FAP, ADA crypt cells and the LI-shift of colon
crypts suggests that formation of ADA crypts, and fully developed
adenomatous polyps, results from an increase in Lz^{0} value caused
by APC mutation during CRC development. An increase in Lz^{0} will

(i) Reduce the maturation of colon crypt cells,

(ii) Increase the fraction of proliferative cells along crypt axis,and

(iii) Increase probability of formation of adenomatous polyps and CRC. Thus, latenzzeit regulatory mechanisms appear to be essential for crypt maintenance and, when altered, contribute to development of CRC.

We acknowledge financial support by Hamline University and advice of Professor Wojicech Komornicki.

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