Department of Applied Mathematics and Statistics, University of California, Santa Cruz, CA, USA

Department of Biology, University of Bergen, Bergen, Norway

Mathematical Ecology Research Group, Department of Zoology, University of Oxford and St Peter’s College, Oxford, UK

Abstract

The hematopoietic stem cell (HSC) system is a demand control system, with the demand coming from the organism, since the products of the common myeloid and lymphoid progenitor (CMP, CLP respectively) cells are essential for activity and defense against disease. We show how ideas from population biology (combining population dynamics and evolutionary considerations) can illuminate the feedback control of the HSC system by the fully differentiated products, which has recently been verified experimentally. We develop models for the penultimate differentiation of HSC Multipotent Progenitors (MPPs) into CLP and CMP and introduce two concepts from population biology into stem cell biology. The first concept is the Multipotent Progenitor Commitment Response (MPCR) which is the probability that a multipotent progenitor cell follows a CLP route rather than a CMP route. The second concept is the link between the MPCR and a measure of Darwinian fitness associated with organismal performance and the levels of differentiated lymphoid and myeloid cells. We show that many MPCRs are consistent with homeostasis, but that they will lead to different dynamics of cells and signals following a wound or injury and thus have different consequences for Darwinian fitness. We show how coupling considerations of life history to dynamics of the HSC system and its products allows one to compute the selective pressures on cellular processes. We discuss ways that this framework can be used and extended.

Introduction

Hematopoiesis (the formation of blood components) is a highly orchestrated and dynamical process. Hematopoietic Stem Cells (HSCs) give rise, through a large array of successively differentiated progeny, to mature blood cells. While progress has been made in understanding the HSC system, particularly at the molecular level

We will show that these questions can only be fully answered if one considers the connection between the needs of the organism and the HSC system, a demand control system

Thus, organismal performance is intimately connected to HSCs and their products

In this paper, we show that the questions raised by Tan et al.

**Details of the HSC model derivation.**

Click here for file

A diagrammatic derivation of Eqns 1 to 6 (details given in Additional file

**A diagrammatic derivation of Eqns** 1 **to** 6 **(details given in Additional file****). ****a)** In the most general case, we consider stem cells (S), a series of Multipotent Progenitor Cells (MPP), a Common Lymphoid Progenitor (CLP) and a Common Myeloid Progenitor (CMP). CLPs give rise to B, NK, and T cells; CMPs give rise to Erythrocytes (E), Granulocytes (G), and Platelets (P). We denote the total numbers of lymphoid and myeloid cells by L and M respectively, rates of differentiation by _{·}(with subscript indicating the cell type involved), rates of development of MPP cells by _{·}, feedback from fully differentiated cells on those rates by _{·}, and rates of cell death by _{·}. The feedback functions have the property that they are 1 when stem cell or fully differentiated cell numbers are low and decline as stem cells or fully differentiated cells increase. Thus, for example, stem cells renew (one stem cell becomes two) at rate _{s}_{s}(_{p}_{s}(_{s}. Similar interpretations hold for other transitions. The Multipotent Commitment Response (MPCR), denoted by **b)** To focus on the MPCR, we combine all of the fully differentiated cells into lymphoid and myeloid classes (L and M) and use Michaelis-Menten-like arguments to compress the MPP class into a single stage, assuming that steady states of intermediate stages are rapidly reached, characterized by combination of rate constants _{N}.

Methods

We begin first by describing, in summary here, with details in Additional file

Dynamics of stem cells and their descendants

In the Additional file

The dynamics are described by the following set of coupled ordinary differential equation:

where _{s}([_{p}([^{′}), and the activity of MPP cells (_{N}[_{l∗}characterizes the additional mortality when the immune system is activated and _{a>b} is an indicator function that is 1 if

**
Symbol
**

**
Interpretation
**

**
Value
**

(Parameter values are a canoncial fixed set, arbitrarily chosen, to illustrate the general principles of an MPCR).

Non-dimensional time

1-3500

[

Concentration of stem cells at time

Eqn 1

[

Concentration of Multipotent Progenitor (MPP) cells at time

Eqn 2

[

Concentration of Common Lymphoid Progenitor (CLP) cells at time

Eqn 3

[

Concentration of Common Myeloid Progenitor (CMP) cells at time

Eqn 4

[

Concentration of fully differentiated Lymphoid (L) cells at time

Eqn 5

[

Concentration of fully differentiated Myeloid (M) cells at time

Eqn 6

Maximum number of stem cells in a niche

10

_{
s
}

Maximum rate of stem cell self-renewal

2.5

Maximum rate of stem cell asymmetrical division

0.001

Feedback control from fully differentiated cells to asymmetric division

Eqn 9

_{s}([

Feedback control from fully differentiated cells to stem cell self-reneval

Eqn 7

_{p}([

Feedback control from fully differentiated cells to symmetric division

Eqn 8

_{s}

Rate of stem cell death

0.004

Rate of MPP multiplication

0.25

_{p}

Rate of MPP cell death

0.02

_{N}

Combination of intermediate multipotent progenitor rate constants

1.0

_{CLP}

Rate of division of CLP into fully differentiated lymphoid cells

0.01

_{CLP}

Rate of CLP cell death

0.001

_{l}

Rate of multiplication of lymphoid cells

0.025

_{l}

Rate of lymphoid cell death when immune system is not activated

0.028

_{l∗}

Additional rate of lymphoid cell death when immune system is activated

0.01

_{a>b}

Indicator function for the inequality

=1 if

_{th}

Threshold concentration for pathogens to activate the immune system

0.025

_{CMP}

Rate of division of CMP into fully differentiated myeloid cells

0.01

_{CMP}

Rate of CMP cell death

0.001

_{m}

Rate of multiplication of myeloid cells

0.0

_{m}

Rate of myeloid cell death

0.01

Value of [L]

varies

Value of [M]

varies

_{sl}(

Feedback control of fully differentiated lymphoid cells on stem cell activity

Eqn 10

_{sm}(

Feedback control of fully differentiated myeloid cells on stem cell activity

Similar to Eqn 10

_{pl}(

Feedback control of fully differentiated lymphoid cells on symmetric renewal

Eqn 11

_{pm}(

Feedback control of fully differentiated myeloid cells on symmetric renewal

Similar to Eqn 10

Feedback control of fully differentiated lymphoid cells on asymmetric renewal

Eqn 11

Feedback control of fully differentiated myeloid cells on asymmetric renewal

Similar to Eqn 10

_{sl}

Feedback parameter in _{sl}(

10

_{pl}

Feedback parameter in _{pl}(

100

Feedback parameter in

20

_{sm}

Feedback parameter in _{sm}(

0.1

_{pm}

Feedback parameter in _{pm}(

0.001

Feedback parameter in

0.2

Coefficient in MPP Commitment Response (MPCR)

Varies

Exponent in MPCR

Varies

_{l}

Density of lymphoid cells in homeostasis

30

_{m}

Density of myeloid cells in homeostasis

30,000

_{h}

Fraction of lymphoid cells in homeostasis

Eqn 15

Rate of accumulation of fitness when myeloid cell concentration is [

Eqn 17

Fitness accumulated to time

Eqn 18

Ratio of organismal to cellular time scale

0.05

Survival to time

Eqn 21

_{e}([

Total rate of mortality when myeloid cell concentration is [

Eqn 20

_{e0}

Myeloid independent rate of mortality

0.05

_{e1}

Myeloid dependent rate of mortality

5.0

_{i}([

Additional rate of mortality when concentration of infectious agents is [

Eqn 20

_{i0}

Coefficient of [

0.02

_{i1}

Coefficient of [^{2} in additional mortality

0.002

_{v}

Replication rate of infectious agents

0.05

_{l}

Clearance rate of infectious agents by lymphoid cells

0.05

_{m}

Clearance rate of infectious agents by myeloid cells

0

_{0}

Concentration of infectious agents at the start of an infection

1

_{th}

Concentration of infectious agents below which additional lymphoid mortality does not occur

0.025

Feedback control, which requires nonlinear dynamics, is essential for the growth and regeneration of tissues. A recent model of genetic products

To incorporate feedbacks we extend Lander et al.’s

where _{sl}(0) = _{sm}(0) = 1(representing the feedback of lymphoid and myeloid cells on stem cell activity), etc, and all _{ij} are decreasing functions of their arguments, as in

where

are parameters. A similar form is used for the feedback control from myeloid cells. This is the simplest form of feedback between the whole organism and the bone marrow stem cell system; see

In a demand control system the probability of a MPP cell ultimately differentiating into a CLP or CMP cell must depend upon the state of the organism. That is, the current densities of myeloid and lymphoid cells determine the appropriate response. We choose a functional form that is widely used in population biology and similar to Michaelis-Menten enzyme kinetics;

where _{m}:_{l} denote the ratio of myeloid to lymphoid cells in homoeostasis. If _{h}denotes the value of

and from Eqn 13 we have

We view the unknowns in this equation as the two parameters

Eqn 16 determines a curve in the _{h}. However, when out of homeostasis, the value of

a) The relationship between the parameters ** α**and

**a) The relationship between the parameters ****and ****of the stem cell commitment response when homeostasis corresponds to 1 lymphoid cell per 1000 myeloid cells. ****b)** Different values of

How an organism goes out of homeostasis depends upon its environment. For example, in an environment when wounds occur frequently, we anticipate the

In this paper, we are interested in

The components of fitness and their dynamics

The representation of genes in subsequent generations is determined by survival and successful reproduction of the focal organism. Regarding the latter, we assume that the rate at which successful reproduction occurs (

provided this expression is positive; otherwise we set _{0}=−1.034565, _{1}=0.001527, and _{2}=−0.0000002864. The peak of ^{∗}=2666(Figure

We assume that the rate at which successful reproduction accumulates, ** f**(

**We assume that the rate at which successful reproduction accumulates, **
**
Δ
**

We let

To determine survival, we assume that uninfected individuals have a per unit time rate of mortality with myeloid-independent and myeloid-dependent components so that the total rate of mortality is (Figure

We assume that the rate of mortality declines with increasing numbers of myeloid cells, which has the effect that annual survival increases with increasing densities of myeloid cells; here we artificially hold the myeloid cells constant

**We assume that the rate of mortality declines with increasing numbers of myeloid cells, which has the effect that annual survival increases with increasing densities of myeloid cells; here we artificially hold the myeloid cells constant.**

Although we focus on non-fatal diseases here, such diseases can still increase mortality rate, e.g. by reducing the effectiveness of flight responses. Hence we assume that the additional mortality induced by the pathogen is

so that

Finally, we incorporate the dynamics of infectious agents. We assume that in the absence of immune response the growth of the infectious agent is exponential with rate _{v} and that lymphoid and myeloid cells clear the infection at rate _{l}and _{m}respectively. In this model, again for simplicity, we ignore memory in the immune system. Thus, if

Integration of Eqns 17-22 forward in time, conditioned on

Eqns 1-22 are a set of deterministic ordinary differential equations that link the behavior of the stem cell system with the needs of the organism. However, organisms in nature experience wounding and infection in a quasi-random manner. We account for this in the following way. Imagine that there are _{w} and _{i}times at which wounds or infections can occur (these values could, of course, be random variables but we treat them as fixed in this paper, only for purposes of simplicity) and then determine a sequence of times _{w}(_{w}),_{w}=1,2,_{w} and _{i}(_{i}),_{i}=1,2,_{i} at which either a wound or infection occurs (in principle both could occur at one time). To illustrate the ideas, we assume that when a wound occurs, myeloid cells drop by 40% and that when an infection occurs, the infectious agent increases to the level _{0}. These occur instantaneously and we then continue with the solution of the differential equations. For the results shown here, we assume that _{w}=_{i}=7 and that the times are uniformly distributed over the interval between day 0 and day 1500.

Results

As introduced above (Figure _{h}ratio of fully differentiated myeloid to lymphoid cells specifies a curve in the _{h} but as illustrated in Figure

To explore this hypothesis, we assume that the rate at which the organism accumulates fitness,

In a ‘deterministic’ or laboratory environment with neither wounding nor infection, organisms still die, so that survival declines with age (Figure

Even in a laboratory environment, without wounding or infection, organism do not live forever, so that survival declines with age (panel a) with the consequence that accumulated fitness saturates

**Even in a laboratory environment, without wounding or infection, organism do not live forever, so that survival declines with age (panel a) with the consequence that accumulated fitness saturates.**

Plotting lifetime accumulated fitness as a function of ** γ**allows us to understand the strength of selection on

**Plotting lifetime accumulated fitness as a function of **
**
γ
**

The alternative to varying

Ten realizations of the model with both wounds and infection, for the case of **=****2**

**Ten realizations of the model with both wounds and infection, for the case of **
**
γ
**

Discussion

Here we have developed and analysed a theoretical framework for linking the population biology of the hematopoietic stem cells to the demands of the individual. We have introduced the notion of the MPCR (Multipotent Progenitor Commitment Response) as the response that the describes the penultimate decision of stem cells before commitment to either a myeloid or lymphoid lineage. We use this response to investigate the control dynamics of a hematopoietic stem cell system and show that different values of the ‘shape’ parameters that describe the MPCR give a range of optimal response. Below we discuss the implications of this on the evolutionary dynamics of the HSC system and, more broadly, for developing a theory of stem cell systems based in population biology.

The meaning of a flat fitness surface

The first derivative,

In previous work

Recently, Huang

How this model can be extended

There are a number of extensions that go beyond the current work for linking population biology and stem cell systems. For instance, experimental validation of the MPCR would require repeated cell count measures of long-term HSCs, short-term HSCs and a range of HSC-derived products. Given such experimental data we envisage that it would be possible to assess goodness of fit between an MPCR model and data (and also characterize unexplained heterogeneties) using computational and statistical methods (Bonsall and Mangel, unpublished).

Furthermore, our framework could be extended to study the consequences of transplant or perturbation effects A perturbation experiment can be modeled by starting the HSC system in its steady state and then reducing the number of lymphoid or myeloid cells and then integrating Eqns 1-22 forward. In addition to predicting the kinetics of fully differentiated cells, we can predict the activity of the stem cells following the perturbation. As described in the Additional file

Computing

We began with the assumption that

Our results point to organismal fitness being a function of fully differentiated lymphoid and myeloid cells, _{l}and _{m} that maximizes organismal fitness.

To do this, we let _{i}(_{i}(

Such a method allows us to represent the optimal production of CLP and CMP cells given that the organism is not infected at time

is the optimal production of MPP cells at time

In the stationary state, fitness is only a function of

This equation provides an explicit form for

Connection to empirical studies

Our work complements the rapid recent development of understanding how gene products that regulate HSCs operate

Thus, currently much is known about the mechanism of HSC system and its descendants

One common way for approaching an understanding of the dynamics of HSCs and their products is through irradiation and transplant experiments (e.g.,

A less commonly used approach, but equally important, is a perturbation experiment in which an animal is challenged in a way that reduces its complement of erythrocytes, platelets or granulocytes (commom myeloid progenitor (CMP) descendants) or lymphocytes (common lymphoid progenitor (CLP) descendants) and HSC activity is observed subsequent to the perturbation. For example, Cheshier et al.

Conclusions

The use of quantitative models to understand the HSC system can be traced to the classic work of Till et al.

In conclusion, our work raises the critical question of how we connect the MPCR with the vast understanding on how signalling shapes HSC products. For instance, appreciating how the MPCR links to T-cell specification

While the broad evolutionary ecological of HSC activity such as mounting an immune response to infection are well-known to have a cost on reproductive fitness (e.g.,

Although raised almost two decades ago

Competing interests

The authors have declared that no competing interests exist.

Authors’ contributions

The model was conceived and designed jointly (MM, MBB). MM analyzed the model and developed the results. The paper was wrote jointly (MM, MBB). Both authors read and approved the final manuscript.

Acknowledgements

We thank Camilla Forsberg, for many discussions on these topics, in which she endeavored to help us understand how the experiments are done, and Oliver Cinquin, German Encisco, Arthur Lander, John Lowengrub, and Edward Nelson for other very helpful conversations. The work of MM is supported by NSF grant EF-0924195.