A Population Balance Model Describing the Cell ... - Wiley Online Library

Biotechnol. Prog. 2007, 23, 1198−1209

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A Population Balance Model Describing the Cell Cycle Dynamics of Myeloma Cell Cultivation Yuan-Hua Liu,† Jing-Xiu Bi,‡ An-Ping Zeng,§ and Jing-Qi Yuan*,† Department of Automation, Shanghai Jiao Tong University, 800 Dongchuan Rd., 200240 Shanghai, PR China, National Key Laboratory of Biochemical Engineering, Institute of Process Engineering, Chinese Academy of Science, 100080 Beijing, PR China, Institute of Bioprocess and Biosystems Engineering, Hamburg University of Technology (TUHH), Denickestrasse 15, D-21073 Hamburg, Germany, and Helmholtz Zentrum fuer Infektionsforschung, Mascheroder Weg 1, D-38124 Braunschweig, Germany

A multi-staged population balance model is proposed to describe the cell cycle dynamics of myeloma cell cultivation. In this model, the cell cycle is divided into three stages, i.e., G1, S, and G2M phases. Both DNA content and cell volume are used to differentiate each cell from other cells of the population. The probabilities of transition from G1 to S and division of G2M are assumed to be dependent on cell volume, and transition probability from S to G2M is determined by DNA content. The model can be used to simulate the dynamics of DNA content and cell volume distributions, phase fractions, and substrate and byproduct concentrations, as well as cell densities. Measurements from myeloma cell cultivations, especially the FACS data with respect to DNA distribution and cell fractions in different stages, are employed for model validation.

1. Introduction In mammalian cell culture, individual cells exhibit heterogeneity as a result of differences in their cellular metabolism and cell-cycle dynamics. The development of modeling techniques that accounts for heterogeneities present at the singlecell level are required (1-4). Population balance model is the most rigorous approach for describing the effects of cell heterogeneities on cell culture dynamics (5, 6). Population balance models are commonly based on a single external state such as cell age (7, 8) or cell mass (5, 9, 10) due to their simplicity. However, age-structured models are of limited practical value because it is difficult to measure the cell age. Contrary to that, some cell properties, e.g., cell volume, DNA content, protein content, etc., are possible to be measured, even at the single-cell level. Therefore, the use of such properties in the formulation of population balance models is more meaningful. Population balance models are assorted in single- and multistaged models depending on the number of cell cycle phases that are included in the formulation (5). There are various reasons that multi-staged models are preferred for an accurate description and control of a cell growth system. Important cellular landmarks, such as DNA replication, occur within a characteristic period of a cell cycle, and therefore some kind of temporal structure is required in population growth models. This can be accomplished by considering the growth within each phase separately and providing appropriate boundary conditions for phase to phase transitions in a multi-staged model (5, 11). Furthermore, there are different biosynthesis capacities at different stages of the cell cycle. For example, maximum secre* To whom correspondence should be addressed. Tel + Fax: +86-213420-4055. E-mail: [email protected]. † Shanghai Jiao Tong University. ‡ Chinese Academy of Science. § Hamburg University of Technology, Helmholtz Zentrum fuer Infectionsforschung. 10.1021/bp070152z CCC: $37.00

tion rate of antibody had been found in the G2M phase of murine hybridoma cells (12). Experimental data derived from Saccharomyces cereVisiae had also shown that a heterologous protein of interest was mainly secreted during the late G2 and M phases (13, 14). Thus, the accurate mathematical description of such processes and the control of the cell cycle specific products can only be achieved by using multi-staged models (5). The intrinsic physiological functions, including growth rate function, transition and division rate functions, and partitioning probability function, are essential parts in population balance models (15). The evolution of flow cytometry (16, 17) has contributed significantly to obtaining single-cell level information. Based on such information, several attempts have been made to determine the intrinsic physiological functions experimentally (18-22). Population balance equations are nonlinear integro-partial differential equations, which are characterized by considerable mathematical complexity (3). Mantzaris and his colleagues have presented several numerical algorithms that can solve these complicated equations rapidly, accurately, and generally (23-25). As stated above, population balance models are powerful for describing the population distribution dynamics of a heterogeneous system. Furthermore, the mathematical formulations of them have been completed to a considerable extent (5, 26, 27). However, intracellular species, such as DNA and protein, are seldom incorporated in the models for applications. Therefore, the objective of this paper is to develop a population balance model with DNA content as the index of cell state for myeloma cell cultivation. The cell cycle is partitioned into three stages by the landmark of DNA replication, i.e., G1, S, and G2M. Apart from DNA content, cell volume is also formulated as an indicator of cell state in the population balance equations. The model is then used to simulate the evolution of DNA content and cell volume distributions as well as cell fraction in each phase. In addition, the dynamics of cell growth, nutrient uptake, and byproduct formation can also be described by such a model.

© 2007 American Chemical Society and American Institute of Chemical Engineers Published on Web 08/11/2007

Biotechnol. Prog., 2007, Vol. 23, No. 5

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The cell cycle measurements by flow cytometry system, i.e., DNA distributions and phase fractions, together with the measurements related to cell growth and cell metabolism from myeloma cell cultivations are used to validate the model.

2. Materials and Methods 2.1. Cell Line and Culture Medium. The myeloma cell line X63-Ag8.653 without recombinant products expression was used as model cell line. The base medium used in flask and spinner cultures was RPMI 1640 supplemented with 10% fetal calf serum (FCS) (Gibco, U.K.), 2 mM glutamine, and 80 µg mL-1 gentamicin. Two feeding media, FM1 and FM2, were used, which had the following compositions: FM1 contained 18 g L-1 glucose and 3 g L-1 glutamine with 10xRPMI; FM2 contained 17.9 g L-1 glucose and 3.2 g L-1 glutamine with 1xRPMI. 2.2. Inoculation and Seed Preparation. Cells stored in liquid nitrogen were thawed and then subcultured five times in T-flasks and spinners. In the subcultures, temperature was maintained at 37 °C, and concentration of CO2 and the humidity were controlled at 5% and 80%, respectively. Exponentially growing precultures were transferred to two 250-mL spinners for further experiments with the initial viable cell density of 1.5 × 105 cells mL-1. 2.3. Cultivation Procedures. All experiments started with a working volume of 150 mL, and glucose and glutamine concentrations of 1.75 and 0.5 g L-1, respectively. Experiment 1: Conventional batch culture. Experiment 2: Batch culture with three pulse feedings. Three times of medium feedings were carried out at 54.5, 73, and 113 h, respectively. At 54.5 h, 2.5 mL FCS and 3.5 mL FM2 were added to achieve the glucose concentration of 1.0 g L-1. At 73 h, 8 mL FCS and 4 mL FM2 were added with the target glucose concentration of 1.0 g L-1. At 113 h, 10 mL FCS and 3.6 mL FM1 were added. Before feeding, 13.6 mL supernatant was removed from the culture medium. Both cell culture experiments were stopped at 134 h. During the cultivation, the temperature was maintained at 37 °C and the rotation rate at 120 rpm, while the concentrations of CO2 and humidity were controlled at 5% and 80%, respectively. 2.4. Analysis Methods. Cell density was determined by microscopic counting with a hemacytometer, and dead cells were evaluated by the trypan blue exclusion method. Glucose and lactate were measured with a YSI 2700 analyzer (Yellow Spring, OH). Ammonia concentration was determined using enzymebased assay kits. Amino acids were analyzed by HPLC (KONTRON, Germany) and evaluated with the Kroma2000 software. DNA content was measured using BD FACS flow cytometer. Cell cycle analysis of the DNA histograms was performed with software Modfit LT (Verity Software House, Inc.). The S-phase fraction was modeled as multiple broadened trapezoids.

3. Population Balance Model 3.1. Cell Cycle Structure. The mammalian cell cycle is conventionally divided into four stages: G1, S, G2, and M phases (28). Most cellular components are synthesized continuously throughout a cell’s life span, whereas DNA synthesis occurs only during a limited portion of the period between cell divisions known as the S phase (29). Thus, the cell cycle may be divided into three parts, G1, S, and G2M phases, by the onset and end of the S phase; see Figure 1. Furthermore, the fractions of the cells in these three parts can be measured experimentally. Therefore, it is practically useful to develop a three-stage model

Figure 1. Cell cycle of myeloma cell line.

for the growth of a mammalian cell population. In this model, both cell volume and DNA content are considered as indicators of the cell state. Figure 1 shows the cell cycle structure. Cell volume accumulates throughout the whole cell cycle. It is postulated that cells in G1 phase are not ready to enter into S phase unless they reach a certain volume (30-33). Similarly, a critical cell volume is also required for cell division in G2M phase. S to G2M transition is assumed to be dependent on the level of DNA content. Cell death is assumed to occur ubiquitously. 3.2. Population Balance Equations. The population balance equation stems from a dynamic cell number balance that includes single cell descriptions of cellular growth and division (34). Derived from the model of Fredrickson et al. (35), the population balance equations in the three-stage model are presented as

∂NG1(V,t) ∂(rV,G1(V,s)NG1(V,t)) + ) ∂t ∂V 2

∫VV

max

PG2M(V,V′)ΓG2M(V′,s)NG2M(V′,t)dV′ -

ΓG1(V,s)NG1(V,t) - rd,G1(V,s)NG1(V,t) -

F(t) N (V,t) (1) V(t) G1

∂NS(V,x,t) ∂(rV,S(V,s)NS(V,x,t)) ∂(rx,S(V,s)NS(V,x,t)) + + ) ∂t ∂V ∂x F(t) -rd,S(V,s)NS(V,x,t) N (V,x,t) (2) V(t) S ∂NG2M(V,t) ∂(rV,G2M(V,s)NG2M(V,t)) + ) ∂t ∂V NS(V,2x1,t) - ΓG2M(V,s)NG2M(V,t) - rd,G2M(V,s)NG2M(V,t) F(t) N (V,t) (3) V(t) G2M where V is the relative cell volume, x is the relative DNA content, s is the environmental state vector, and t is the culture time. The domain of definition is Vmin e V e Vmax, x1 e x e 2x1, and t > 0, where, Vmin and Vmax are the minimum and maximum cell volumes, respectively; x1 denotes the relative DNA content of G-phase cells. NΩ(V,t), where Ω denotes G1 or G2M, is the cell number density function, and NΩ(V,t) dV represents the number of Ω-phase cells per unit bio-volume that at time t have volume between V and V + dV. NS(V,x,t) is defined as a joint cell number density function so that NS(V,x,t) dV dx is the number of S-phase cells per unit bio-volume that at time t have volume between V and V + dV and DNA content between x and x + dx. rV,G1, rV,S, and rV,G2M are the increase rates of the single-cell volume in G1, S, and G2M, respectively. rx,S is the DNA synthesis rate in S phase. rd,G1, rd,S, and rd,G2M are the cell death rates in G1, S, and G2M, respectively. ΓG1 is the transition rate from G1 to S. ΓG2M is the division rate of G2M-


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phase cells. PG2M(V,V′) is the partitioning probability density function, which represents the probability that a mother cell in G2M phase with volume V′ will give birth to a daughter cell of volume V. F is the feed flow rate. V is the culture volume. Equation 1 gives the population balance equation of G1 phase. The terms in the equation are accumulation, growth, source, transition, death ,and dilution in sequence. Growth term accounts for the loss of cells due to the fact that they grow into bigger cells. Source term represents the rate of birth of cells originating from the division of all bigger cells in G2M phase. The factor 2 multiplying the integral birth term arises from the fact that each division event leads to the generation of two daughter cells. The transition term accounts for the loss of cells due to transition to the next phase. The death term is to express the loss of cells due to cell death. The dilution term represents the dilution of cell number density caused by feeding. The population balance equation of S phase is depicted with eq 2. The loss of cells stemming from DNA synthesis is expressed with the third term of the left-hand side. Cells transited from G1 phase are located on the boundary between G1 phase and S phase where cells have DNA content of x1. Therefore, there is no source term in this equation. Neither does the transition term appear in the equation because the transition of cells from S to G2M is deterministic. The last two terms are similar to those described in G1 phase. The number densities in cell volume space (NV,S) and DNA content space (Nx,S) may be found from the joint number density NS(V, x, t) in the two spaces by integration:

NV,S(V,t) )

∫x

Nx,S(x,t) )

∫VV

2x1 1

max

min

NS(V,x,t) dx

(4)

NS(V,x,t) dV

(5)

Equation 3 shows the population balance equation of G2M phase. The first term right-handed denotes the source cells derived from S phase. The second one is the division term that expresses the loss of cells due to cell division. Other terms are similar to those of eq 1. 3.3. Boundary Conditions. For the solution of eqs 1-3, appropriate boundary conditions are required. By using the law of mass and number conservations at cell cycle transitions, Fredrickson had derived the boundary conditions for a general multi-staged population balance model in detail (27). Accordingly, the boundary conditions in this model are obtained as follows. (1) The boundary condition of G1 phase:

NG1(Vmin,t) ) 0

(6)

(2) The boundary conditions of S phase:

NS(Vmin,x,t) ) 0

(7)

rV,SNS(V,x1,t) ) ΓG1NG1(V,t)

(8)

(3) The boundary condition of G2M phase:

NG2M(Vmin,t) ) 0

(9)

3.4. Intrinsic Physiological Functions. The rates by which each cell grows, consumes substrate, and forms products will, in general, depend on the cell volume as well as the environmental circumstance (26). The same is also true for the rate of cell transition or division. Glutamine was found to be the

Figure 2. Diagram of the regulator model combined with Monod model for the single-cell volume increase rate and DNA synthesis rate.

limiting substrate in our experiments. In addition, other essential amino acids were observed depleted during the cultivations. Among these essential amino acids, lysine was observed to be consumed rapidly and exhausted earlier than others. For the purpose of simplicity, lysine limitation is regarded to represent the lumped effect of all amino acid limitation. As found by Simpson et al. (36), there existed a short-term growth after deprivation of any single amino acid. The amino acid demand during such a short-term growth period should be supplied by an intracellular amino acid pool (37, 38). Accordingly, intracellular lysine is finally regarded as limiting substrate. Under these assumptions, the single-cell volume increase rates of the cells in G1 and G2M phases, as well as the DNA synthesis rate of S-phase cells, are first modeled with Monod kinetics:

r′w,Ω ) kΩ

Lysint,Ω Gln for phases Ω ) Gln + KGln,Ω Lysint,Ω + KLys,Ω

{

G1 and G2M, w ) V (10) S, w)x

where Gln is the glutamine concentration. Lysint,Ω is the intracellular level of lysine of Ω-phase cells. kΩ is the maximum single-cell volume increase rate or maximum DNA synthesis rate. KGln,Ω and KLys,Ω are saturation constants. Obviously, the Monod-type eq 10 cannot account for the lag period after inoculation or after feeding in batch culture with pulse feedings. To deal with the lag phase, a metabolic regulator model, proposed by Bellgardt (39) and well validated (39-41), is applied; see Figure 2. Such a regulator model is proposed to mimic the induction of the enzyme pool involved in metabolic network. The dashed frame outlines a first-order regulator with rw,Ω as input and rR,Ω as output. rmin represents the minimum rate dependent on the constitutive activity of the network. k2 is a constant in analogy to the degradation rate coefficient. The introduction of the negative feedback by k3rw,Ω is to account for the dilution of the enzyme pool due to cell growth (42). The switch K is a low pass switch, namely, rw,Ω ) min{r′w,Ω,rR,Ω}. Mathematically, the regulator model is depicted with eq 11. The real increase rate of single-cell volume or DNA synthesis rate rw,Ω is obtained from eq 12. Since DNA synthesis is accompanied by cell volume increment (43), the increase rate of single-cell volume in S phase is assumed proportional to DNA synthesis rate, with the ratio of kV/x, namely, rV,S ) kV/xrx,S.

drR,Ω ) k1,Ωrw,Ω - (k2 + k3rw,Ω)rR,Ω + rmin dt

(11)

rw,Ω ) min(r′w,Ω , rR,Ω)

(12)

Nutrient limitation is one of the primary causes of cell death during the decline phase of batch culture (44). Observations in


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Table 1. Parameter Values Obtained Independent of Model Fitting rmin (h-1) 0.0001

k2

k3

µG1

µS

µG2M

σG1

σS

0.01 0.0001 1.408 1.650 1.920 0.10 0.05

σG2M

q

0.08

40

our experiments showed that the limitation of amino acids especially lysine may be responsible for cell death. Therefore, the death rates of the cells in G1, S and G2M phases are assumed determined by intracellular lysine:

rd,Ω ) kd

Kd Lysint,Ω + Kd

for phases Ω ) G1,S,G2M

(13)

where kd and Kd are constants. Furthermore, for G1 and G2M phases, the respective transition and division rates are modeled according to the results found in the literature (15, 45):

ΓΩ(V,s) )

fΩ(V) 1-

∫0V fΩ(V′) dV′

where fΩ(V) is the transition probability density function, which is assumed to be dependent on cell volume. Experimental data demonstrated that fΩ(V) can be approximated with Gaussian distribution (46), with the mean value µΩ and the standard deviation σΩ. The partitioning probability density function PG2M(V,V′) is assumed as a β distribution type, see eq 15, which was also applied in other studies (5, 11, 47).

1 1 V q-1 V q-1 1V′ β(q,q) V′ V′

() (

)

(15)

3.5. Mass Balances on the Substrates and Byproducts. The mass balances of glucose, glutamine, and extracellular lysine are given below:

ds F(t) ) (s - s) dt V(t) F Vmax Vmax 1 1 rV,G1NG1(V,t) dV r N (V,t) dV V V min YG1/s YS/s min V,S V,S Vmax 1 r N (V,t) dV (16) YG2M/s Vmin V,G2M G2M

∫

(

F(t) Vmax dLac 1 )r N (V,t)dV + Lac + YLac/Glc V dt YG1/Glc min V,G1 G1 V(t) Vmax 1 r N (V,t)dV + YS/Glc Vmin V,S V,S Vmax 1 r N (V,t)dV (17) YG2M/Glc Vmin V,G2M G2M

∫

∫

)

∫

where Lac is the lactate concentration. YLac/Glc represents the lactate yield on glucose. Lactate is assumed to be produced at a rate proportional to the uptake rate of glucose. Intracellular lysine is assumed constant before extracellular lysine depletion (37, 48, 49), namely, Lysint,Ω ) Lysint0(constant). When extracellular lysine is depleted, the mass balances of intracellular lysine of the cells in G1, S, and G2M phases are described as

d(Lysint,ΩV) dV )for phases Ω ) G1,S,G2M dt YΩ/Lys dt (18)

rV,Ω(V,s) for phases Ω ) G1,G2M (14)

PG2M(V,V′) )

The mass balance of the byproduct (lactate) is

∫

∫

where s represents Glc, Gln, or Lys; Glc is the concentration of glucose; sF is the concentration of substrate s in the feeding medium; YG1/s, YS/s, and YG2M/s are the cell volume yields on substrate s of G1, S, and G2M, respectively. The three integral terms in eq 16 are the substrate consumption rates leading to cell growth in G1, S, and G2M phases, respectively.

After calculating the derivative of the function Lysint,ΩV, eq 18 becomes

dLysint,Ω 1 dV Lysint,Ω dV )dt VYΩ/Lys dt V dt

(19)

Because dV/dt ) rV,Ω, the dynamics of the intracellular lysine concentration finally is

dLysint,Ω Lysint,Ω 1 )rV,Ω r for phases Ω ) dt V YΩ/LysV V,Ω G1,S,G2M (20) The first term in eq 20 is to account for the dilution of the intracellular substrate due to cell growth. The second term is the intracellular substrate uptake leading to cell growth. 3.6. Simulations. The initial distributions of cell volume and DNA content must be provided for eqs 1-3. According to the initial DNA distribution from experimental data, cells were uniformly distributed with respect to DNA content in S phase. In order to estimate the initial cell volume distribution, it is postulated first that there is no cell death and all cells divide upon reaching the volume 2V0. At the same time, a cell’s growth rate is assumed independent of its volume. Then, the initial cell volume distribution may be depicted as f(V) ) f(V0)2(1-V/V0), since the precultures were obtained from steady exponentially growing cultures (50). Given the initial total cell density N0, f(V0) ) 2 ln2N0/V0 is obtained. Furthermore, the initial cell fraction of each phase can be calculated as follows:

FG1(0) )

1 N0

∫VV

max,G1

f(V) dV

(21)

0

Table 2. Parameter Values Identified by Model Fitting kG1 (h-1) 0.198 YG1/Gln (g-1) 20 YLac/Glc 0.85

kS (h-1) 0.077 YS/Gln (g-1) 45 kV/x 0.35

kG2M (h-1) 0.13 YG2M/Gln (g-1) 30 k1,G1 1.3

KLys,G1 (g L-1) 0.15 YG1/Glc (g-1) 8 k1,S 1.0

KLys.S (g L-1) 0.1 YS/Glc (g-1) 16 k1,G2M 1.1

KLys,G2M (g L-1) 0.1 YG2M/Glc (g-1) 9

KGln,G1 (g L-1) 0.6 YG1/Lys (g-1) 400

KGln,S (g L-1) 0.05 YS/Lys (g-1) 800

KGln,G2M (g L-1) 0.05 YG2M/Lys (g-1) 500

kd (h-1) 0.05 Kd (g-1) 0.06


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Figure 3. Time courses of Expt 1: (a) concentrations of glutamine and lysine, (b) concentrations of glucose and lactate, (c) fractions of the cells in G1, S, and G2M phases, and (d) viable and dead cell densities. Scatters: experimental measurements. Lines: simulation results.

FS(0) )

1 N0

FG2M(0) )

∫VV

max,S

f(V) dV

(22)

max,G1

1 N0

∫V

2V0

f(V) dV

(23)

max,S

where Vmax,G1 and Vmax,S are the maximum volumes of G1and S-phase cells, respectively. Since experimental results showed that FG1(0) ) 0.5, FS(0) ) 0.32, and FG2M(0) ) 0.18, then Vmax,G1 ) 1.42V0 and Vmax,S ) 1.76V0 can be obtained. In practice, randomness occurs during the processes of cell growth, division, and partitioning of the mother cell into two daughter cells. Therefore, the cell volume may be smaller (or larger) than V0 (or 2V0), and there is no fixed boundary between two neighbor phases. Thus, as defined above, the domain of cell volume is [Vmin, Vmax], where Vmin < V0 and Vmax > 2V0. The cell volume distribution of each phase should be broadened by Gaussian function. Thus, the initial conditions are

NG1(V,0) )

∫V

1

1.42V0 0

x2π θV,G1

4 ln2N0 V0 x2π θV,G1

∫V1.42V 0

0

(

exp -

(

exp -

)

(V - V′)2 2θV,G12 2

(V - V′) 2θV,G12

)

f(V′) dV′ )

2-V′/V0 dV′ (24)

NS(V,x,0) )

4 ln2N0

0

V0 x2π θV,Sx1

NG2M(V,0) ) 4 ln2N0 V0 x2π θV,G2M

1.76V ∫1.42V

2V ∫1.76V

0

(

exp -

0

0

(

exp -

)

(V - V′)2 2

2θV,S

(25)

)

(V - V′)2 2θV,G2M2

2-V′/V0 dV′

2-V′/V0 dV′ (26)

According to experimental data reported in the literature (51), it is assumed that θV,G1 ) θV,S ) θV,G2M ) 0.05V0. Furthermore, experimental results showed that the values of DNA content in G1 and G2M phases are not exactly x1 and 2x1, respectively. This may arises from experimental error and the fact that cells are not illuminated evenly during flow cytometry (52-54). The domain of DNA content is therefore extended to be between xmin and xmax, where xmin < x1 and xmax > 2x1. Thus, the equations with DNA content involved must be verified. The population balance eq 2, the individual number density in cell volume space eq 4, the boundary condition eq 8, and the initial distribution eq 25 for S phase should be replaced by eqs 27, 30, 31, and 32, respectively.

∂NS(V,x,t) ∂(rV,S(V,s)NS(V,x,t)) ∂(rx,S(V,s)NS(V,x,t)) + + ) ∂t ∂V ∂x PS(x)ΓG1(V,s)NG1(V,t) - ΓS(x,s)NS(V,x,t) F(t) rd,S(V,s)NS(V,x,t) N (V,x,t) (27) V(t) S


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Figure 4. Time courses of Expt 2: (a) concentrations of glutamine and lysine, (b) concentrations of glucose and lactate, (c) cell fractions in G1, S and G2M phases, and (d) viable and dead cell densities (arrows indicate feedings). Scatters: experimental measurements. Lines: simulation results.

with

PS(x) )

1

e-(x-x1) /2θx,S 2

x2π θx,S

2

(28)

and

fS(x)

ΓS(x,s) )

1-

NV,S(V,t) )

∫0x fS(x′) dx′ ∫xx

max

rx,S(V,s)

NS(V,x,t) dx

min

(30)

NS(V,xmin,t) ) 0 NS(V,x,0) )

∫xx

max

min

(

exp -

(x - x′) 2θx,S

2

)

2

dx′

(29)

(31)

4 ln2N0 2πV0θx,SθV,Sx1

1.76V ∫1.42V

0

0

(

exp -

)

(V - V′)2 2θV,S2

2-V′/V0 dV′ (32)

where PS is defined as distributing probability density function,

Figure 5. Expt 1.

Evolution of DNA content density distribution of

which is assumed to be a Gaussian function with the mean value x1 and standard deviation θx,S ) 0.05x1. That is, the cells transited from G1 to S will be distributed around the very point x ) x1 instead of being located at that point. ΓS is the transition rate from S to G2M. fS(x) is the transition probability density function, which is taken to be a Gaussian function with the mean value and standard deviation of µS and σS, respectively. The population balance eq 3 for G2M phase is substituted by eq 33:

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Figure 6. DNA distributions of Expt 1 at 0, 12, 44, 70, and 90 h. Left panels: FACS data. Right panels: simulation results.


1205

∂NG2M(V,t) ∂(rV,G2M(V,s)NG2M(V,t)) + ) ∂t ∂V

∫xx

max

min

ΓS(x,s)NS(V,x,t) dx - ΓG2M(V,s)NG2M(V,t) rd,G2M(s)NG2M(V,t) -

F(t) N (V,t) (33) V(t) G2M

Total viable cell density NV is

NV(t) )

∫VV

max

min

(NG1(V,t) + NV,S(V,t) + NG1(V,t)) dV (34)

The dynamics of the dead cell density is

dND(t) ) dt

∫VV

(rd,G1NG1(V,t) + rd,SNV,S(V,t) +

max

min

rd,G2MNG2M(V,t)) dV (35)

Figure 7. Evolution of DNA content density distribution of Expt 2.

Cell fractions in G1, S, and G2M phases are calculated as follows:

a function that has the same distribution type with m(x,t), namely, C(x,t), is structured. At the same time, the scale of x should be in accordance with the measurements and the integral of C(x,t) in x space must be NExp(t). Thus,

FG1(t) )

1 NV

∫VV

NG1(V,t) dV

(36)

FS(t) )

1 NV

∫VV

NV,S(V,t) dV

(37)

1 FG2M(t) ) NV

∫V

NG2M(V,t) dV

max

min

max

min

Vmax min

(38)

DNA distributions in G1 phase and G2M phase are obtained by broadening the total cells in the respective G1 and G2M phases:

MG1(x,t) )

1

x2π θx,G1

MG2M(x,t) ) 1

x2π θx,G2M

(

(

exp -

exp -

)∫

(x - x1)2 2

2θx,G1

)∫

(x - 2x1)2 2θx,G2M

2

Vmax

Vmin

Vmax

Vmin

NG1(V,t) dV (39)

NG2M(V,t) dV (40)

where MG1 and MG2M are defined as the DNA distribution functions of G1 and G2M phases, respectively. According to experimental results, θx,G1 and θx,G2M are 0.05x1 and 0.1x1, respectively. Then, the DNA density distribution function of the whole cell cycle is

m(x,t) )

MG1(x,t) + Nx,S(x,t) + MG2M(x,t)

∫x

xmax min

(MG1(x,t) + Nx,S(x,t) + MG2M(x,t)) dx

(41)

The cell volume density distribution function of the whole cell cycle is

n(V,t) )

NG1(V,t) + NV,S(V,t) + NG2M(V,t)

∫V

Vmax min

(NG1(V,t) + NV,S(V,t) + NG2M(V,t)) dV

(42)

In addition, the counts for sample (NExp) and mean DNA content of G1 (x1,Exp) varied with culture time t (see Figures 7 and 8). In order to compare the model simulations of DNA distribution with the measurements in a straightforward manner,

C(x,t) ) m(x/B(t),t)NExp(t)/B(t)

(43)

where B(t) ) x1,Exp(t)/x1. C(x,t) is then used to describe the DNA distribution of the cells for sample at time t. An implicit finite difference numerical method is employed to solve the nonlinear integro-partial differential equations. We choose a time-step 4t ) 0.01, cell volume grid 4V ) 0.01, and DNA content grid 4x ) 0.01. Both x1 and V0 are set as a value of 1. The minimum and maximum values of cell volume and DNA content are set as Vmin ) xmin ) 0, Vmax ) xmax ) 3.

4. Results and Discussion The model is validated with a rigorous batch culture and a batch culture with three pulse feedings for a myeloma cell line. The data collected in these experiments included DNA content distributions, phase fractions, cell densities, and the concentrations of glutamine, glucose, lysine, and lactate. The constant intracellular lysine concentration before extracellular lysine depletion (Lysint0) is assumed to be 0.1 g L-1, about four times the extracellular lysine level (37, 48, 49). Some of the model parameters can be obtained before simulation; see Table 1. rmin, k2, and k3 in the regulator model are estimated according to previous studies associated with enzyme kinetics (42, 55). µS and σS are determined by analyzing the DNA distribution from our measurements. µG1, σG1, µG2M, σG2M, and q are estimated according to the previous work (5, 23, 46). Other parameters must be identified by tuning the model simulations to measurements; see Table 2. Model simulations of the cell fraction in each phase, the densities of viable and dead cells, and the concentrations of glutamine, glucose, lysine, and lactate are compared with measurements of Expts 1 and 2 in Figures 3 and 4, respectively. Evolutions of DNA content distribution of Expts 1 and 2 are simulated and provided with Figures 5 and 6, respectively. Profiles of DNA distribution at 0, 12, 44, 70, and 90 h of Expt 1 are presented in Figure 7. Figure 8 gives the profiles of DNA distribution at 22, 60, 76, 113, and 125 h of Expt 2. DNA distribution profiles from simulation results at all these time points qualitatively agree with that from measurements. Quantitatively, obvious deviation may be found at 12 and 90 h in Expt 1.

1206


Figure 8. DNA distributions of Expt 2 at 22, 60, 76, 113, and 125 h. Left panels: FACS data. Right panels: simulation results.

Both simulation results and measurements show that fluctuations of S-phase and G1-phase fractions occurred after inoculation or after feeding; see Figures 3c and 4c. After inoculation,

cells first experienced a lag phase (10 h or so) and then entered a rapid growth phase. During the lag phase, more and more G1-phase cells entered into S phase and stayed there, as can be


1207

Figure 9. Evolution of cell volume density distribution of Expt 1.

Figure 10. Evolution of cell volume density distribution of Expt 2.

found in the profile of DNA distribution at 12 h in Figure 7. This resulted in an increment of S-phase fraction and a decrement of G1-phase fraction. In general, fractions of each phase will finally reach a steady state during the rapid growth period (56, 57). However, this phenomenon was not observed in our experiments because the duration of the exponential phase is too short for cells to reach a steady state. As cells entered the reduced growth period, G1-phase cell growth becomes slower, whereas there’s no apparent variation in S phase, as coincides with the results of previous studies (58-61). Thus, cells accumulated in G1 phase. In Expt 2, feedings at 54.5, 73, and 113 h had also caused similar fluctuations of S-phase fraction and G1-phase fraction as that after inoculation. But these fluctuations are slighter. The fraction of G2M phase was relatively constant during the cultivations, although slight fluctuation was observed. The consumption of intracellular amino acids cannot be ignored. The profiles of phase fractions of Expt 1, see Figure 3c, showed that S-phase fraction kept decreasing and G1-phase fraction kept increasing during the period between 60 and 100 h. This indicates that some S-phase cells went through the rest cell cycle and entered into G1 phase by division during that period. However, experimental results showed that some essential amino acids especially lysine in the medium had been depleted before 60 h; see Figure 3a. Therefore, the amino acids requirement for cell growth and DNA synthesis during that period must be derived from the intracellular pool. This model can also be used to simulate the evolution of the cell volume distribution. Figures 9 and 10 provide the cell volume density distributions of Expt 1 and Expt 2, respectively.

description for single cell growth is required for further investigations. For example, cybernetic approach will be a suitable alternative because it takes account of the intrinsic physiological information and it is competent to describe the dynamic responses after inoculation or after feeding without any additional metabolic regulator model (62, 63). Studies on cell cycle regulation have revealed that cells are driven through the cell cycle by a series of (Cdks) engines, consisting of a catalytic kinase subunit and a regulatory cyclin subunit (64). In mammalian cell lines, there are many kinds of Cdk-cyclin complexes, which trigger the cell-cycle events (65). The kinetics of these complexes has been modeled in different ways (66, 67). The approach proposed in this work is capable of incorporating such temporal cellular structure and accounting for events that control the cell division cycle. To further improve the model, it would be desirable to replace cell volume by other cell-cycle-specific parameters, such as proteins (Cdks) or metabolites (8). We are working at TUHH on metabolic analysis of animal cell culture. There might be some possibilities.

5. Concluding Remarks The framework of a three-stage population balance model presented here provides an approach to describe the dynamics of DNA distribution and phase fractions during mammalian cell cultivations. Both cell volume and DNA content are formulated in the population balance equations as indicators of cell state, making it possible to predict the evolution of cell volume and DNA content distributions. Qualitatively, comparison between simulation results and measurements shows that the model can describe the myeloma cell population dynamics. However, the quantitative disagreements occur in the profiles of phase fractions and DNA distributions. This may be mainly because the Monod-type kinetics of the increase rate of single-cell volume or DNA synthesis rate is too simple to account for the sophisticated intrinsic physiological behaviors. A structured

Notation x, V x1 V0 xmin, xmax, Vmin, Vmax x1,Exp B Ω w NG1, NS, NG2M NV,S, Nx,S MG1, MG2M m, n C ΓG1, ΓS, ΓG2M fG1, fS, fG2M

relative DNA content and cell volume relative DNA content in G1 phase relative cell volume at birth under ideal condition minimum or maximum relative DNA content and cell volume mean DNA content of G1 phase from measurements ratio function of x1,Exp to x1 cell cycle phase (G1,S or G2M) cellular states (x or V) cell number density functions of G1, S and G2M phases (×108 cells L-1) individual cell number density functions of S phase in cell volume and DNA content spaces (×108 cells L-1) DNA distribution functions of G1 and G2M phases (×108 cells L-1) DNA content and cell volume density distribution functions DNA distribution function corresponding to experimental results (cells) transition rate functions of G1 and S phases and division rate function of G2M phase (cells h-1) transition probability density functions of G1 and S phases and division probability density function of G2M phase


1208 PG2M PS µΩ, σΩ

partitioning probability density function of G2M phase distributing probability density function of S phase mean value and standard deviation of transition or division probability density function θx,Ω, θV,Ω standard deviations of broadening functions rw,Ω real increase rate of cellular state w of Ω-phase cells (h-1) r′w,Ω, increase rate of Ω-phase cells from Monod model and rR,Ω regulator model (h-1) rd,Ω single cell death rate of Ω-phase (h-1) Glc, Gln, concentrations of glucose, glutamine and lysine (g L-1) Lys s environmental state vector s, sF substrate concentration in the culture and feeding medium (g L-1) Lysint,Ω intracellular levels of lysine of Ω-phase cells (g) constant intracellular lysine before extracellular lysine Lysint0 depletion (g) Lac lactate concentration (g L-1) YΩ/Glc, cell volume yields on glucose, glutamine and lysine of YΩ/Gln, Ω phase (g-1) YΩ/Lys lactate yield on glucose (g g-1) YLac/Glc kG1, kS, maximum single-cell volume increase rates or DNA kG2M synthesis rate (h-1) KGln,Ω, Monod constants of single-cell volume increase rate or KLys,Ω DNA synthesis rate (g L-1) ratio between the cell volume increase rate and DNA kV/x synthesis rate kd, Kd single cell death constants k1,Ω, k2, constants of the regulator k3, rmin sample counts for cell cycle analysis (cells) NExp F flow rate of feeding (L h-1) V culture volume (L)

Acknowledgment The authors gratefully acknowledge the financial support of the Natural Science Foundation of China (Grant nos. 60574038 and 20576136), the German DAAD Postdoc Scholarship (J.X.B.), and the DFG/Germany-MOE/China Exchange Program (J.-Q.Y.). GBF (Gesellschaft fuer Biotechnologische Forschung GmbH (now called Helmholtz Zentrum fu¨r Infektionsforschung) is acknowledged for providing all experimental and analysis facilities. We sincerely thank Mrs. Angela Walter/GBF and Mr. Joachim Hammer/GBF for analytical assistance.

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