Medical Hypotheses M&d HypdhM m92) 39,191~194 0 L.oqpnan CroupUK Ltd 1992
Precipitous Stem Cell (SC) Differentiation and 1ff Noise G.T. MATlOLl USC School of Medicine, 2011 Zonal Avenue, Los Angeles, CA 90033, USA
Abstract - After a brief digression on certain types of noise of generic interest, it is concluded that l/f noise offers interesting interpretations for unusual modes of SC differentiation such as the precipitous maturation of a subset of the macrophage lineage. Functionally equivalent to weak, spurious sources these specialized cells support the proliferation and maturation of erythroid cells extending thus the stimulatory influence of true, strong sources although in a rather indirect fashion.
Introduction Certain aspects of SC renewal and decline to terminal differentiation are interpretable in the context of a probability space-field where p (the renewal probability) is distributed multimodally along a decaying gradient having the stem state bounded by 0.5 5 p 5 1. The probability field extends to some ‘chemical’ distance from a strong source which feeds to parasitic and directly coupled SCs a stimnlus P for mitotic renewal (1,2,3). However, SC proliferation is a synergistic process requiting certain growth prefactors (Prefs) aud specialized differentiating factors (Diis) additional to P (4). These ancillary signals cooperate in a complex way to trigger numerous mitotic cycles, yielding substantial numbers of mature cells. In this scenario, SCs behave as ‘excitable systems’ with the following phases. 1.
The recovery phase is the period during which SCs replenish their store of P through direct source contacts.
Date received 16 January 1992 Date accepted 10 February 1992
2.
During the inert phase primitive @ = 1) SCs in particular are refractory to mitogenic factors (such as Difs) unless prirmd by Pmfs. In the hemopoietic system the priming action could be exercised by one of several ‘let&ins eliciting no or very few mitoses before the SC enters the ‘excited’ phase.
3.
In the excited phase SCs are fully responsive to Difs. Generally this phase is characterized by sustained mitotic activity required for the segregation of one or several differentiation options executed along trajectorial paths traced by combinatorial concentrations of Difs, encountered by SCs or by theirprogeny at distinct field locations. Such a hierarchical cascade mirrors the decay of p to zero as the SC progeny differentiates through a succession of programmatic steps of variable duration.
The algebra capturing important aspects of such processes is encapsulated by the Fokker-Plank (I?) 191
hEDICAL HYPo+lnEsEs
192 equation: y
= $[
fb)+I$]p(x,t)
Eq.1
namely: the progeny of @ = 1) SCs drifts to certain probability values @ < 1) depending upon the spatial distribution of Difs at given field locations as well as upon some ‘diffusion-like’ process, alluded to by the second right-hand operator. Because of the relationship between FP and Langevin equations (2), it is permissible to reinterpret diffusion in terms of noise so that D becomes the intensity. Equation 1 says that any change in SC renewal would be entirely deterministic except for the unavoidable occurrence of some kind of noise. (As an aside, we remark that this is the correct definition of stochasticity. Regrettably.abuse of tee has identified stochasticitywith randomness, causing needless confusion in various disciplines.) In sum, albeit stochastic processes are ultimately understood in statistical terms, their history and evolution are not random Noise does not always interfere with determinism in a destructive sense. Assume, for example, that a differentiationtrajectory and branchesthereof run into a seemingly unsurmountable block due to few missing signals, inactive genes, etc. In such cases noise might rescue abortive differentiation along one or multiple detours (whenever accessible) or it may force the trajectory around or over some barrier after its height (width) has been lowered (narrowed)however irregularly so. Likewise, noise can induce a (stem) cell to complete a differentiation program by sampling a heterogeneous spectrum of alternative trajectories incompletely sketched by suboptimal combinationsof Difs.
(v is the infinitesimal volume confining noise to a point-like event of vanishing width.) Colored noise 6) inserts concatenated actions into the dynamics. f is exponentially correlated within certain time windows as: q
0) f tub =%exp(--
t-t’
tc )
where tc is the correlation interval. f may play a role particularlywhen cell-cell and gene-gene interactions attempt to set up correlated behaviour amongst momentarily inde ndent substates. Finally, the l! iJeor flicker noise relates to tiactal time-space properties of any SC field. The states of SC renewal and differentiation are controlled by a number N(t) of temporally fluctuating para.nWers such as source strength, local concentration of Pmfs and Difs, clustered to special reaction surfaces such as expanding (contracting)extracellular matrices, depletion layers, etc. The power spectrum (viz: the square modulus of the Fourier transform of these flue ations) is related to the flicker noise as &(f) - l/ P , with exponent fl defined below. For p = 1,the autocorrelationfunction decays logarithmically. In our context, all this means is that the most pluripotent (p = 1) SC should decay to p values below 0.5 less abruptly than other oligo-or bipotent SCs. Similarly the most immature but committed cells should mitose extensively before reaching terminal maturation @ = 0) through many short time steps (s) rather than via very steep or rapid declines (S) requiring fewer divisions. Defining fi (= log s/log S) as the fiactal declin of p, a typical SC sample should decay as: exp(-kt) 5 for 0 < p 4. This behaviour, known as the Kohlraush stretched exponential relaxation, means that in any hierarchically stmctured system (such as typical SC fields) there is always a A short digression on noisyprocesses number of individuals (cells) that relax with time dependencies significantly distinct from the population. Having the dimension of fractal plicative, additive, intrinsic (or molecular), extrinsic, time, fJis related to the l/f noise. etc), we briefly discuss the white [?$t)J, the colored Additional interest in the l/f noise stems from con{E(t)]and the flicker (l/f, fractal) in particular, due to sidering the frequency of precipitously catastrophic its importance for peculiar modes of SC differ- events in general. For example, massive earthquakes entiation, discussed below. capable of levelling an entire town are less likely than The white noise (11)modulates SC dynamics via those of average magnitude causing limited damage. Gaussianperturbations of: This is so whenever the probability of a catastrophic event E, is the product of many independent lesser
Precipitous Stem Cell (SC) Differentiation and 1ff Noise G.T. MATlOLl USC School of Medicine, 2011 Zonal Avenue, Los Angeles, CA 90033, USA
Abstract - After a brief digression on certain types of noise of generic interest, it is concluded that l/f noise offers interesting interpretations for unusual modes of SC differentiation such as the precipitous maturation of a subset of the macrophage lineage. Functionally equivalent to weak, spurious sources these specialized cells support the proliferation and maturation of erythroid cells extending thus the stimulatory influence of true, strong sources although in a rather indirect fashion.
Introduction Certain aspects of SC renewal and decline to terminal differentiation are interpretable in the context of a probability space-field where p (the renewal probability) is distributed multimodally along a decaying gradient having the stem state bounded by 0.5 5 p 5 1. The probability field extends to some ‘chemical’ distance from a strong source which feeds to parasitic and directly coupled SCs a stimnlus P for mitotic renewal (1,2,3). However, SC proliferation is a synergistic process requiting certain growth prefactors (Prefs) aud specialized differentiating factors (Diis) additional to P (4). These ancillary signals cooperate in a complex way to trigger numerous mitotic cycles, yielding substantial numbers of mature cells. In this scenario, SCs behave as ‘excitable systems’ with the following phases. 1.
The recovery phase is the period during which SCs replenish their store of P through direct source contacts.
Date received 16 January 1992 Date accepted 10 February 1992
2.
During the inert phase primitive @ = 1) SCs in particular are refractory to mitogenic factors (such as Difs) unless prirmd by Pmfs. In the hemopoietic system the priming action could be exercised by one of several ‘let&ins eliciting no or very few mitoses before the SC enters the ‘excited’ phase.
3.
In the excited phase SCs are fully responsive to Difs. Generally this phase is characterized by sustained mitotic activity required for the segregation of one or several differentiation options executed along trajectorial paths traced by combinatorial concentrations of Difs, encountered by SCs or by theirprogeny at distinct field locations. Such a hierarchical cascade mirrors the decay of p to zero as the SC progeny differentiates through a succession of programmatic steps of variable duration.
The algebra capturing important aspects of such processes is encapsulated by the Fokker-Plank (I?) 191
hEDICAL HYPo+lnEsEs
192 equation: y
= $[
fb)+I$]p(x,t)
Eq.1
namely: the progeny of @ = 1) SCs drifts to certain probability values @ < 1) depending upon the spatial distribution of Difs at given field locations as well as upon some ‘diffusion-like’ process, alluded to by the second right-hand operator. Because of the relationship between FP and Langevin equations (2), it is permissible to reinterpret diffusion in terms of noise so that D becomes the intensity. Equation 1 says that any change in SC renewal would be entirely deterministic except for the unavoidable occurrence of some kind of noise. (As an aside, we remark that this is the correct definition of stochasticity. Regrettably.abuse of tee has identified stochasticitywith randomness, causing needless confusion in various disciplines.) In sum, albeit stochastic processes are ultimately understood in statistical terms, their history and evolution are not random Noise does not always interfere with determinism in a destructive sense. Assume, for example, that a differentiationtrajectory and branchesthereof run into a seemingly unsurmountable block due to few missing signals, inactive genes, etc. In such cases noise might rescue abortive differentiation along one or multiple detours (whenever accessible) or it may force the trajectory around or over some barrier after its height (width) has been lowered (narrowed)however irregularly so. Likewise, noise can induce a (stem) cell to complete a differentiation program by sampling a heterogeneous spectrum of alternative trajectories incompletely sketched by suboptimal combinationsof Difs.
(v is the infinitesimal volume confining noise to a point-like event of vanishing width.) Colored noise 6) inserts concatenated actions into the dynamics. f is exponentially correlated within certain time windows as: q
0) f tub =%exp(--
t-t’
tc )
where tc is the correlation interval. f may play a role particularlywhen cell-cell and gene-gene interactions attempt to set up correlated behaviour amongst momentarily inde ndent substates. Finally, the l! iJeor flicker noise relates to tiactal time-space properties of any SC field. The states of SC renewal and differentiation are controlled by a number N(t) of temporally fluctuating para.nWers such as source strength, local concentration of Pmfs and Difs, clustered to special reaction surfaces such as expanding (contracting)extracellular matrices, depletion layers, etc. The power spectrum (viz: the square modulus of the Fourier transform of these flue ations) is related to the flicker noise as &(f) - l/ P , with exponent fl defined below. For p = 1,the autocorrelationfunction decays logarithmically. In our context, all this means is that the most pluripotent (p = 1) SC should decay to p values below 0.5 less abruptly than other oligo-or bipotent SCs. Similarly the most immature but committed cells should mitose extensively before reaching terminal maturation @ = 0) through many short time steps (s) rather than via very steep or rapid declines (S) requiring fewer divisions. Defining fi (= log s/log S) as the fiactal declin of p, a typical SC sample should decay as: exp(-kt) 5 for 0 < p 4. This behaviour, known as the Kohlraush stretched exponential relaxation, means that in any hierarchically stmctured system (such as typical SC fields) there is always a A short digression on noisyprocesses number of individuals (cells) that relax with time dependencies significantly distinct from the population. Having the dimension of fractal plicative, additive, intrinsic (or molecular), extrinsic, time, fJis related to the l/f noise. etc), we briefly discuss the white [?$t)J, the colored Additional interest in the l/f noise stems from con{E(t)]and the flicker (l/f, fractal) in particular, due to sidering the frequency of precipitously catastrophic its importance for peculiar modes of SC differ- events in general. For example, massive earthquakes entiation, discussed below. capable of levelling an entire town are less likely than The white noise (11)modulates SC dynamics via those of average magnitude causing limited damage. Gaussianperturbations of: This is so whenever the probability of a catastrophic event E, is the product of many independent lesser
