6.9 Mechanical Feedback
305
Perhaps the most hotly debated item in the above list has been the second
hypothesis. Some researchers contend that stress is the appropriate growth stimulus,
while others support strain. As discussed earlier in this section, since growth is a
tensor, it is likely that the stimulus also is a tensor quantity. In this respect, either
stress or strain will do, and arguments can be made supporting each.
A popular argument favoring strain is that cells possess known tools for sensing
deformation directly, whereas force must be sensed indirectly. For instance, during
mechanotransduction, stretching of cell-cell and cell-matrix adhesions can open
folded proteins, allowing other proteins to bind in the open slots (Eyckmans et al.
2011; Zhu 2014). This can trigger a chain of molecular events, i.e., a signaling
cascade, that enhances or inhibits growth. Similarly, as discussed in Chap. 1,
stretching cell membranes can open ion channels (Zhu 2014), and forces transmitted
through the cytoskeleton can deform the nucleus, changing the conformation of
DNA and altering gene expression (Fedorchak et al. 2014; Szczesny and Mauck
2017).
Other considerations support stress as the growth stimulus. Some argue that,
unlike strain, stress (actually Cauchy stress) depends only on the current configuration and does not require a reference configuration that may be relevant only at
some time in the distant past, e.g., during development. On the other hand, some
problems have a relatively well-defined reference state. The heart, for example,
normally returns to nearly the same end-diastolic geometry with each beat, and
changes in sarcomere length provide a convenient measure for strain. Thus, strain
may be an appropriate choice for the heart (Kerckhoffs et al. 2012). Alternatively,
strain rate combines the advantages of stress and strain, as its reference is the current
configuration. However, strain rate has seldom been used in formulating mechanical
growth laws.
Observations on muscle growth provide evidence favoring stress (at least for
contractile tissue). It is well known that exercise causes muscles to grow larger,
with the greatest growth occurring during isometric exercise (Goldberg et al. 1975),
when force production and stress are highest (see Fig. 5.7c) but strain is essentially
zero. To address some of the issues with strain, some investigators have proposed
strain-based growth laws in terms of elastic strain or stretch ratio λ ∗ . However, the
reference state for λ ∗ may never be realized in vivo.
Considering the above discussion, we propose the following. At the molecular
level, a growth response depends on the deformation of mechanosensitive proteins
(Eyckmans et al. 2011; Zhu 2014), but deforming these proteins requires a certain
level of force exerted by cells and tissues. Consequently, the growth stimulus could
be strain at the molecular level and stress at the cell and tissue levels.
The basic idea is illustrated by a simple model consisting of bars (representing
stress fibers, actin fibers, or the cell membrane) attached to a mechanosensor
(adhesions, the nucleus, or an ion channel) (Fig. 6.18). Suppose growth depends
on how much the mechanosensor is stretched by the tension F in the bars. In other
words, the growth rate depends on the value of F . Notably, this does not imply that
growth depends explicitly on the strain in the bars, as stiffer bars require less stretch
than more compliant bars to deliver the same force (Fig. 6.18a). Thus, in contrast to a
305
Perhaps the most hotly debated item in the above list has been the second
hypothesis. Some researchers contend that stress is the appropriate growth stimulus,
while others support strain. As discussed earlier in this section, since growth is a
tensor, it is likely that the stimulus also is a tensor quantity. In this respect, either
stress or strain will do, and arguments can be made supporting each.
A popular argument favoring strain is that cells possess known tools for sensing
deformation directly, whereas force must be sensed indirectly. For instance, during
mechanotransduction, stretching of cell-cell and cell-matrix adhesions can open
folded proteins, allowing other proteins to bind in the open slots (Eyckmans et al.
2011; Zhu 2014). This can trigger a chain of molecular events, i.e., a signaling
cascade, that enhances or inhibits growth. Similarly, as discussed in Chap. 1,
stretching cell membranes can open ion channels (Zhu 2014), and forces transmitted
through the cytoskeleton can deform the nucleus, changing the conformation of
DNA and altering gene expression (Fedorchak et al. 2014; Szczesny and Mauck
2017).
Other considerations support stress as the growth stimulus. Some argue that,
unlike strain, stress (actually Cauchy stress) depends only on the current configuration and does not require a reference configuration that may be relevant only at
some time in the distant past, e.g., during development. On the other hand, some
problems have a relatively well-defined reference state. The heart, for example,
normally returns to nearly the same end-diastolic geometry with each beat, and
changes in sarcomere length provide a convenient measure for strain. Thus, strain
may be an appropriate choice for the heart (Kerckhoffs et al. 2012). Alternatively,
strain rate combines the advantages of stress and strain, as its reference is the current
configuration. However, strain rate has seldom been used in formulating mechanical
growth laws.
Observations on muscle growth provide evidence favoring stress (at least for
contractile tissue). It is well known that exercise causes muscles to grow larger,
with the greatest growth occurring during isometric exercise (Goldberg et al. 1975),
when force production and stress are highest (see Fig. 5.7c) but strain is essentially
zero. To address some of the issues with strain, some investigators have proposed
strain-based growth laws in terms of elastic strain or stretch ratio λ ∗ . However, the
reference state for λ ∗ may never be realized in vivo.
Considering the above discussion, we propose the following. At the molecular
level, a growth response depends on the deformation of mechanosensitive proteins
(Eyckmans et al. 2011; Zhu 2014), but deforming these proteins requires a certain
level of force exerted by cells and tissues. Consequently, the growth stimulus could
be strain at the molecular level and stress at the cell and tissue levels.
The basic idea is illustrated by a simple model consisting of bars (representing
stress fibers, actin fibers, or the cell membrane) attached to a mechanosensor
(adhesions, the nucleus, or an ion channel) (Fig. 6.18). Suppose growth depends
on how much the mechanosensor is stretched by the tension F in the bars. In other
words, the growth rate depends on the value of F . Notably, this does not imply that
growth depends explicitly on the strain in the bars, as stiffer bars require less stretch
than more compliant bars to deliver the same force (Fig. 6.18a). Thus, in contrast to a